Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04980946pt" id="M1" height="9.34695pt" version="1.1" viewBox="-0.0498162 -9.29714 12.5971 9.34695" width="12.5971pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path id="g198-7" d="M896 662C863 643 826 624 783 624C682 624 539 680 429 680C328 680 253 659 202 616C163 583 138 537 138 484C138 399 208 349 283 349C396 349 466 417 492 554L472 555C436 442 385 379 286 379C234 379 179 410 179 491C179 578 257 634 382 634C475 634 622 574 731 574C796 574 862 610 907 647L896 662ZM657 568C577 499 520 440 477 341L463 310C368 277 286 233 247 186L260 171C320 211 380 237 437 256C349 77 281 31 184 31C124 31 83 59 78 94H80C87 83 101 81 109 81C137 81 153 104 153 131S130 180 103 180C68 180 43 154 43 111C43 34 123 0 192 0C335 0 469 75 546 282C599 291 640 293 661 294C653 267 650 244 650 224V206L709 226C706 235 706 245 706 252C706 271 706 289 714 312C758 333 786 368 786 387C786 403 775 409 763 409C744 409 712 391 685 352C644 351 604 345 564 337C595 433 622 506 671 554L657 568Z"/></g></svg> be a family of continuous functions defined on a compact interval. We give a sufficient condition so that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M2" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 42.6009 12.0395" width="42.6009pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g198-7"/></g><g transform="matrix(.0135,0,0,-0.0135,15.479,0)"><path id="g117-59" d="M535 194V548H477V194C477 102 391 35 294 35S111 102 111 194V548H53V194C53 72 166 -15 294 -15S535 72 535 194Z"/></g><g transform="matrix(.0135,0,0,-0.0135,26.46,0)"><path id="g113-124" d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 269V273C199 286 222 334 222 388C222 454 209 523 209 588C209 632 218 671 293 681V709C234 709 148 695 148 577C147 513 155 438 155 372C155 337 152 291 64 285V256C152 250 155 204 155 169C156 105 148 31 148 -41C148 -157 234 -169 293 -169Z"/></g><g transform="matrix(.0135,0,0,-0.0135,31.161,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/></g><g transform="matrix(.0135,0,0,-0.0135,37.664,0)"><path id="g113-126" d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z"/></g></svg> contains a dense <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.29458pt" id="M3" height="6.70873pt" version="1.1" viewBox="-0.0498162 -6.41415 5.33525 6.70873" width="5.33525pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path id="g200-30" d="M279 468H265C237 462 212 452 191 438C134 403 97 369 81 338S57 259 57 196C57 95 89 23 153 -18H167L338 48L329 71C297 54 267 45 239 45C215 45 197 53 185 70C168 93 159 140 159 212V272C159 332 166 371 179 388C183 393 190 399 199 404L260 328L344 384L279 468Z"/></g></svg>-generated free algebra; in other words, <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04980946pt" id="M4" height="9.34695pt" version="1.1" viewBox="-0.0498162 -9.29714 12.5971 9.34695" width="12.5971pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g198-7"/></g></svg> is densely <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.29458pt" id="M5" height="6.70873pt" version="1.1" viewBox="-0.0498162 -6.41415 5.33525 6.70873" width="5.33525pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g200-30"/></g></svg>-strongly algebrable. As an application we obtain dense <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.29458pt" id="M6" height="6.70873pt" version="1.1" viewBox="-0.0498162 -6.41415 5.33525 6.70873" width="5.33525pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g200-30"/></g></svg>-strong algebrability of families of nowhere H&#xf6;lder functions, Bruckner-Garg functions, functions with a dense set of local maxima and local minima, and nowhere monotonous functions differentiable at all but finitely many points. We also study the problem of the existence of large closed algebras within <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M7" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 42.6009 12.0395" width="42.6009pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g198-7"/></g><g transform="matrix(.0135,0,0,-0.0135,15.479,0)"><use xlink:href="#g117-59"/></g><g transform="matrix(.0135,0,0,-0.0135,26.46,0)"><use xlink:href="#g113-124"/></g><g transform="matrix(.0135,0,0,-0.0135,31.161,0)"><use xlink:href="#g113-49"/></g><g transform="matrix(.0135,0,0,-0.0135,37.664,0)"><use xlink:href="#g113-126"/></g></svg> where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.6616993pt" id="M8" height="12.9302pt" version="1.1" viewBox="-0.0498162 -12.2685 42.3797 12.9302" width="42.3797pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g198-7"/></g><g transform="matrix(.0135,0,0,-0.0135,12.451,0)"><path id="g117-77" d="M512 1V51H381C244 51 133 142 133 255S244 459 381 459H512V509H382C213 509 75 395 75 255S213 1 382 1H512Z"/></g><g transform="matrix(.0136,0,0,-0.0136,24.189,0)"><path id="g197-29" d="M712 0L497 285C582 312 628 378 628 466C628 609 518 662 390 662H70V0H236V271H310L505 0H712ZM491 604C551 577 584 532 584 464C584 401 550 345 491 321V604ZM447 320C428 317 411 315 392 315H236V618H397C412 618 432 613 447 610V320ZM626 44H528L363 271C392 271 422 270 451 275L626 44ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.0095,0,0,-0.0095,34.377,-5.984)"><path id="g50-89" d="M782 650H541L536 618L564 614C597 609 604 600 584 574C535 508 486 451 437 395C394 473 361 533 343 575C331 605 335 610 375 615L400 618L407 650H156L147 618C222 612 230 608 267 537L375 330C277 214 186 117 163 96C108 43 97 38 27 32L18 0H265L274 32L251 35C204 41 202 47 224 80C264 138 327 213 397 293L504 80C520 48 516 42 472 36L440 32L433 0H707L716 32C627 38 622 43 581 121L455 361L629 548C685 606 696 610 775 618L782 650Z"/></g></svg> or <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.6616993pt" id="M9" height="12.9302pt" version="1.1" viewBox="-0.0498162 -12.2685 41.8085 12.9302" width="41.8085pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g198-7"/></g><g transform="matrix(.0135,0,0,-0.0135,12.451,0)"><use xlink:href="#g117-77"/></g><g transform="matrix(.0136,0,0,-0.0136,24.189,0)"><path id="g197-14" d="M632 159C554 79 496 30 378 30C330 30 251 50 213 80V583C258 614 324 632 379 632C497 632 552 583 630 503L661 534C584 627 498 676 375 676C172 676 45 529 45 332C45 140 176 -14 374 -14C497 -14 586 35 663 128L632 159ZM169 118C160 125 155 134 148 145C111 201 89 264 89 332C89 416 122 490 169 558V118Z"/></g><g transform="matrix(.0095,0,0,-0.0095,33.806,-5.984)"><use xlink:href="#g50-89"/></g></svg>. We prove that the set of perfectly everywhere surjective functions together with the zero function contains a <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M10" height="10.5382pt" version="1.1" viewBox="-0.0498162 -10.4884 10.7902 10.5382" width="10.7902pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.0095,0,0,-0.0095,6.503,-5.984)"><use xlink:href="#g200-30"/></g></svg>-generated algebra closed in the topology of uniform convergence while it does not contain a nontrivial algebra closed in the pointwise convergence topology. We prove that an infinitely generated algebra which is closed in the pointwise convergence topology needs to contain two valued functions and infinitely valued functions. We give an example of such an algebra; namely, it was shown that there is a subalgebra of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M11" height="12.3849pt" version="1.1" viewBox="-0.0498162 -12.3351 17.8671 12.3849" width="17.8671pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g197-29"/></g><g transform="matrix(.0095,0,0,-0.0095,10.12,-5.984)"><use xlink:href="#g197-29"/></g></svg> with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M12" height="10.5382pt" version="1.1" viewBox="-0.0498162 -10.4884 10.7902 10.5382" width="10.7902pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-51"/></g><g transform="matrix(.0095,0,0,-0.0095,6.503,-5.984)"><use xlink:href="#g200-30"/></g></svg> generators which is closed in the pointwise topology and, for any function <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571899pt" id="M13" height="13.2498pt" version="1.1" viewBox="-0.0498162 -9.6779 8.83017 13.2498" width="8.83017pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"/></g></svg> in this algebra, there is an open set <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M14" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 9.98608 9.21094" width="9.98608pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-86" d="M715 650H476L470 622C550 619 562 610 562 570C562 536 553 481 535 384L523 321C485 121 408 37 283 37C178 37 99 109 134 297L178 532C193 612 199 616 278 622L284 650H29L23 622C105 616 109 609 94 532L48 291C23 160 51 83 98 40C143 -1 202 -16 260 -16C321 -16 387 3 439 49C509 111 542 204 563 312L577 384C595 479 608 533 619 569C629 603 637 620 709 622L715 650Z"/></g></svg> such that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571899pt" id="M15" height="15.65pt" version="1.1" viewBox="-0.0498162 -12.0781 39.0774 15.65" width="39.0774pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-103"/></g><g transform="matrix(.0095,0,0,-0.0095,8.704,-5.984)"><path id="g54-33" d="M556 236V289H56V236H556Z"/></g><g transform="matrix(.0095,0,0,-0.0095,14.499,-5.984)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g><g transform="matrix(.0135,0,0,-0.0135,19.654,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0135,0,0,-0.0135,24.342,0)"><use xlink:href="#g113-86"/></g><g transform="matrix(.0135,0,0,-0.0135,34.195,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> is a Bernstein set.

Artur Bartoszewicz

Szymon G&#x142;&#x105;b

Large Function Algebras with Certain Topological Properties

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M1" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 9.40133 9.21094" width="9.40133pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-72" d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"/></g></svg> be a discrete group and let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2537994pt" id="M2" height="9.46933pt" version="1.1" viewBox="-0.0498162 -9.21553 11.7268 9.46933" width="11.7268pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path id="g198-2" d="M837 672C829 675 822 674 813 674C710 674 608 541 510 398C475 447 422 480 349 480C260 480 162 415 162 299C162 199 212 133 284 92C229 39 191 15 134 15C91 15 69 45 69 65H71C78 54 95 53 100 53C128 53 143 80 143 101C143 128 121 149 93 149C60 149 31 121 31 77C31 24 74 -15 138 -15C198 -15 255 22 311 79C360 56 416 43 473 39L455 0H552L569 42C612 49 661 63 693 87L684 103C657 88 617 76 581 69C677 301 753 477 846 658L837 672ZM772 638C701 530 628 379 559 227C558 279 547 330 526 371C609 494 688 609 771 638H772ZM536 199C536 191 536 184 535 176L486 67C430 71 380 84 335 105C395 173 451 257 507 344C527 301 536 251 536 199ZM492 372C431 277 369 183 310 119C245 158 203 217 203 301S275 447 349 447C416 447 461 417 492 372Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2537899pt" id="M3" height="9.64612pt" version="1.1" viewBox="-0.0498162 -9.39233 13.0187 9.64612" width="13.0187pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path id="g198-3" d="M895 660C863 648 835 636 808 620C736 668 639 687 564 687C395 687 153 589 153 403C153 307 221 260 300 260C447 260 534 401 534 561H514C506 396 415 290 305 290C234 290 194 335 194 396C194 555 383 657 566 657C626 657 716 643 776 601C669 531 592 432 501 271C389 71 313 16 189 16C118 16 69 60 69 95H71C78 84 93 83 101 83C129 83 144 106 144 131C144 160 121 179 93 179C60 179 34 150 34 107C34 41 100 -15 207 -15C365 -15 509 76 621 311C626 300 647 292 668 292C675 292 701 294 717 298C752 278 763 243 763 203C763 140 734 78 685 49C666 38 647 34 624 34C596 34 574 55 574 82C574 127 623 171 679 176L675 199C586 184 537 132 537 67C537 32 567 -11 631 -11C672 -11 710 0 743 22C804 63 839 127 839 186C839 234 813 289 749 316V318C837 329 902 398 902 463C902 524 873 566 830 601C850 615 874 628 902 641L895 660ZM841 485C841 420 802 346 717 338C697 349 675 352 662 352C655 352 644 351 639 348C691 449 736 524 801 580C825 555 841 524 841 485Z"/></g></svg> be two subgroups of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M4" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 9.40133 9.21094" width="9.40133pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-72"/></g></svg>-valued continuous functions defined on two 0-dimensional compact spaces <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M5" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 10.5029 9.04777" width="10.5029pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-89" d="M748 650H522L515 622L546 617C580 611 587 604 565 575C518 513 469 451 419 393C376 474 349 534 330 580C318 609 325 612 361 618L383 622L392 650H151L144 622C214 616 224 612 257 543L360 327C270 218 187 124 159 95C106 40 92 34 26 28L17 0H252L259 28L236 31C189 37 188 47 209 78C249 136 308 210 377 294L478 79C494 44 487 37 449 32L418 28L409 0H673L680 28C596 34 591 39 554 116L436 361C526 469 574 521 604 553C659 612 669 614 739 622L748 650Z"/></g></svg> and
<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M6" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 8.96616 9.04777" width="8.96616pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-90" d="M640 650H413L406 622L438 617C475 611 483 605 463 571C422 503 346 400 296 333C264 403 215 529 199 575C189 604 191 614 228 618L262 622L271 650H30L23 622C90 615 97 609 119 553L212 321C222 295 223 285 218 259L193 127C176 39 167 33 82 28L76 0H350L356 28C266 36 257 40 275 127L303 266C308 290 314 302 328 322C396 418 454 493 498 546C551 610 565 616 631 622L640 650Z"/></g></svg>. A group isomorphism <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M7" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 11.5228 9.04777" width="11.5228pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g></svg> defined between <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2537994pt" id="M8" height="9.46933pt" version="1.1" viewBox="-0.0498162 -9.21553 11.7268 9.46933" width="11.7268pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g198-2"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2537899pt" id="M9" height="9.64612pt" version="1.1" viewBox="-0.0498162 -9.39233 13.0187 9.64612" width="13.0187pt"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><use xlink:href="#g198-3"/></g></svg> is called <i>separating</i> when,
for each pair of maps <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571899pt" id="M10" height="13.2498pt" version="1.1" viewBox="-0.0498162 -9.6779 46.6851 13.2498" width="46.6851pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"/></g><g transform="matrix(.0135,0,0,-0.0135,7.072,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.0135,0,0,-0.0135,12.432,0)"><path id="g113-104" d="M546 430L539 434C529 434 505 438 495 440C473 444 450 448 430 448C352 448 265 412 213 366C145 306 96 203 96 103C96 22 135 -12 160 -12C190 -12 238 14 262 32C310 68 368 120 411 184H413C403 117 396 75 384 21C353 -118 325 -158 291 -184C270 -200 241 -205 208 -205C133 -205 90 -164 74 -110C70 -98 58 -100 49 -107C34 -119 23 -140 23 -155C23 -190 74 -261 166 -261C219 -261 280 -233 314 -208C383 -157 446 -79 470 81C491 223 529 388 546 430ZM456 386C452 357 433 283 420 252C402 216 366 174 325 129C288 88 239 56 212 56C192 56 182 77 182 120C182 165 199 242 226 292C256 348 281 377 311 389C327 395 353 402 375 402C408 402 436 394 456 386Z"/></g><g transform="matrix(.0135,0,0,-0.0135,23.928,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g><g transform="matrix(.0136,0,0,-0.0136,34.866,0)"><use xlink:href="#g198-2"/></g></svg> satisfying that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571899pt" id="M11" height="15.65pt" version="1.1" viewBox="-0.0498162 -12.0781 122.598 15.65" width="122.598pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-103"/></g><g transform="matrix(.0095,0,0,-0.0095,8.704,-5.984)"><path id="g54-33" d="M556 236V289H56V236H556Z"/></g><g transform="matrix(.0095,0,0,-0.0095,14.499,-5.984)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g><g transform="matrix(.0135,0,0,-0.0135,19.654,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0135,0,0,-0.0135,24.342,0)"><path id="g113-102" d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z"/></g><g transform="matrix(.0095,0,0,-0.0095,29.883,3.264)"><path id="g50-72" d="M722 297H451L442 267C561 257 561 253 547 182L531 98C521 44 465 24 390 24C228 24 130 134 130 296C130 482 252 628 465 628C592 628 657 579 656 458L692 461C695 540 699 605 702 631C664 642 590 667 492 667C203 667 24 502 24 285C24 93 170 -17 368 -17C464 -17 543 11 617 28C613 43 612 68 620 106L635 182C649 255 652 258 716 268L722 297Z"/></g><g transform="matrix(.0135,0,0,-0.0135,36.978,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.0135,0,0,-0.0135,44.694,0)"><path id="g117-59" d="M535 194V548H477V194C477 102 391 35 294 35S111 102 111 194V548H53V194C53 72 166 -15 294 -15S535 72 535 194Z"/></g><g transform="matrix(.0135,0,0,-0.0135,55.675,0)"><use xlink:href="#g113-104"/></g><g transform="matrix(.0095,0,0,-0.0095,63.385,-5.984)"><use xlink:href="#g54-33"/></g><g transform="matrix(.0095,0,0,-0.0095,69.18,-5.984)"><use xlink:href="#g50-50"/></g><g transform="matrix(.0135,0,0,-0.0135,74.335,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,79.023,0)"><use xlink:href="#g113-102"/></g><g transform="matrix(.0095,0,0,-0.0095,84.564,3.264)"><use xlink:href="#g50-72"/></g><g transform="matrix(.0135,0,0,-0.0135,91.659,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0135,0,0,-0.0135,100.132,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,111.87,0)"><use xlink:href="#g113-89"/></g></svg>, it holds that
<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.571899pt" id="M12" height="15.65pt" version="1.1" viewBox="-0.0498162 -12.0781 140.508 15.65" width="140.508pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-73"/></g><g transform="matrix(.0135,0,0,-0.0135,9.344,0)"><use xlink:href="#g113-103"/></g><g transform="matrix(.0095,0,0,-0.0095,18.048,-5.984)"><use xlink:href="#g54-33"/></g><g transform="matrix(.0095,0,0,-0.0095,23.843,-5.984)"><use xlink:href="#g50-50"/></g><g transform="matrix(.0135,0,0,-0.0135,28.998,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,33.686,0)"><use xlink:href="#g113-102"/></g><g transform="matrix(.0095,0,0,-0.0095,39.227,3.264)"><use xlink:href="#g50-72"/></g><g transform="matrix(.0135,0,0,-0.0135,46.322,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0135,0,0,-0.0135,54.038,0)"><use xlink:href="#g117-59"/></g><g transform="matrix(.0135,0,0,-0.0135,65.019,0)"><use xlink:href="#g113-73"/></g><g transform="matrix(.0135,0,0,-0.0135,75.043,0)"><use xlink:href="#g113-104"/></g><g transform="matrix(.0095,0,0,-0.0095,82.753,-5.984)"><use xlink:href="#g54-33"/></g><g transform="matrix(.0095,0,0,-0.0095,88.547,-5.984)"><use xlink:href="#g50-50"/></g><g transform="matrix(.0135,0,0,-0.0135,93.703,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,98.391,0)"><use xlink:href="#g113-102"/></g><g transform="matrix(.0095,0,0,-0.0095,103.932,3.264)"><use xlink:href="#g50-72"/></g><g transform="matrix(.0135,0,0,-0.0135,111.027,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.0135,0,0,-0.0135,119.5,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.0135,0,0,-0.0135,131.237,0)"><use xlink:href="#g113-90"/></g></svg>. We prove that under some mild conditions every biseparating isomorphism <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2537899pt" id="M13" height="9.64612pt" version="1.1" viewBox="-0.0498162 -9.39233 71.825 9.64612" width="71.825pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-73"/></g><g transform="matrix(.0135,0,0,-0.0135,15.169,0)"><path id="g113-59" d="M116 412C83 412 59 388 59 355C59 321 83 297 115 297S170 321 170 355C170 388 147 412 116 412ZM114 -12C147 -12 171 11 171 46C171 78 147 103 115 103C84 103 60 78 60 46C60 11 84 -12 114 -12Z"/></g><g transform="matrix(.0136,0,0,-0.0136,22.043,0)"><use xlink:href="#g198-2"/></g><g transform="matrix(.0135,0,0,-0.0135,39.682,0)"><path id="g117-148" d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z"/></g><g transform="matrix(.0136,0,0,-0.0136,58.664,0)"><use xlink:href="#g198-3"/></g></svg> can be represented by means of a continuous function <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129898pt" id="M14" height="9.89089pt" version="1.1" viewBox="-0.0498162 -9.6779 62.1425 9.89089" width="62.1425pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-105" d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z"/></g><g transform="matrix(.0135,0,0,-0.0135,10.776,0)"><use xlink:href="#g113-59"/></g><g transform="matrix(.0135,0,0,-0.0135,17.65,0)"><use xlink:href="#g113-90"/></g><g transform="matrix(.0135,0,0,-0.0135,32.545,0)"><use xlink:href="#g117-148"/></g><g transform="matrix(.0135,0,0,-0.0135,51.526,0)"><use xlink:href="#g113-89"/></g></svg> as a weighted composition operator. As a consequence we establish
the equivalence of two subgroups of continuous functions if there is a biseparating
isomorphism defined between them.

Marita Ferrer

Margarita Gary

Salvador Hern&#xe1;ndez

Representation of Group Isomorphisms: The Compact Case

We deal with a new type of statistical convergence for double sequences, called <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M3" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 9.9589 9.04777" width="9.9589pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-152"/></g></svg>-<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M4" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 9.65971 9.04777" width="9.65971pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-66"/></g></svg>-statistical convergence, and we prove a Korovkin-type approximation theorem with respect to this type of convergence in modular spaces. Finally, we give some application to moment-type operators in Orlicz spaces.

Carlo Bardaro

Antonio Boccuto

Kamil Demirci

Ilaria Mantellini

Sevda Orhan

Korovkin-Type Theorems for Modular <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 9.9589 9.04777" width="9.9589pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-152" d="M699 659C666 654 622 640 595 610C579 591 569 572 562 530L552 472C530 342 504 307 403 302V529C403 604 411 617 483 622V650H238V622C310 617 318 604 318 529V302C232 308 194 347 172 471L161 531C153 573 145 592 131 609C103 643 59 654 25 659L15 632C62 611 63 583 73 524L85 452C97 383 116 347 155 316C199 282 255 269 318 267V122C318 46 310 33 233 28V0H489V28C412 33 403 46 403 122V267C470 269 518 283 559 315C599 346 618 378 632 450L646 519C658 580 665 608 709 632L699 659Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="725" vert-adv-y="725" /></g></svg>-<svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 9.65971 9.04777" width="9.65971pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-66" d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="703" vert-adv-y="703" /></g></svg>-Statistical Convergence

Complex graphics of dynamical system have striking features of fractals and become a wide area of research due to their beauty and complexity of their nature. The aim of this paper is to study dynamics of relative superior tricorns and multicorns using <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" id="M2" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 6.51834 9.21094" width="6.51834pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-84"/></g></svg>-iteration schemes. Several examples are presented to explore the geometry of relative superior tricorns and multicorns for antipolynomial <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3626003pt" id="M3" height="11.1243pt" version="1.1" viewBox="-0.0498162 -10.7617 63.6615 11.1243" width="63.6615pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-123" d="M475 445L452 468C423 436 401 426 385 426S352 427 305 436C268 443 216 448 185 448C156 448 128 425 102 382C89 360 77 338 64 304L89 291C117 340 143 380 190 380C214 380 268 376 295 373C320 370 344 371 362 374L70 101C32 65 23 36 23 18C23 5 27 -3 31 -8C36 -8 39 -6 42 -1C52 15 67 31 80 40S110 49 143 33C195 8 257 -20 290 -20C339 -20 382 11 439 129L416 148C366 74 334 55 305 55C278 55 236 68 203 81C169 94 141 100 122 96C205 170 348 308 411 376L475 445Z"/></g><g transform="matrix(.0135,0,0,-0.0135,12.804,0)"><path id="g117-148" d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z"/></g><rect height="0.679945" width="6.77227" x="31.7853" y="-9.07051"/><g transform="matrix(.0135,0,0,-0.0135,31.785,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.0095,0,0,-0.0095,38.558,-6.371)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g><g transform="matrix(.0135,0,0,-0.0135,46.958,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.0135,0,0,-0.0135,57.938,0)"><path id="g113-100" d="M383 397C383 424 344 448 299 448C244 448 172 409 132 375C66 319 23 227 23 146C23 42 74 -12 146 -12C208 -12 298 30 359 103L343 124C315 95 248 48 192 48C145 48 111 85 111 163C111 228 129 294 151 330C171 363 201 401 241 401C275 401 302 384 325 356C332 347 339 344 348 348C373 360 383 381 383 397Z"/></g></svg> of complex polynomial <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3626003pt" id="M4" height="10.7367pt" version="1.1" viewBox="-0.0498162 -10.3741 31.8037 10.7367" width="31.8037pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.0095,0,0,-0.0095,6.748,-5.984)"><use xlink:href="#g50-111"/></g><g transform="matrix(.0135,0,0,-0.0135,15.148,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.0135,0,0,-0.0135,26.129,0)"><use xlink:href="#g113-100"/></g></svg> for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-1.16493pt" id="M5" height="9.86369pt" version="1.1" viewBox="-0.0498162 -8.69876 29.0065 9.86369" width="29.0065pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g><g transform="matrix(.0135,0,0,-0.0135,10.586,0)"><path id="g117-94" d="M531 285V335L57 550V494L473 310L57 127V71L531 285ZM531 -40V10H57V-40H531Z"/></g><g transform="matrix(.0135,0,0,-0.0135,22.324,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g></svg>.

Shin Min Kang

Arif Rafiq

Abdul Latif

Abdul Aziz Shahid

Young Chel Kwun

Tricorns and Multicorns of <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2129908pt" height="9.21094pt" version="1.1" viewBox="-0.0498162 -8.99795 6.51834 9.21094" width="6.51834pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-84" d="M449 634C442 637 425 643 405 650C376 660 341 666 307 666C181 666 98 590 98 485C98 400 170 343 215 310L246 288C307 243 343 204 343 147C343 67 291 18 219 18C104 18 61 124 51 202L23 199C28 124 27 71 27 47C47 22 122 -16 204 -16C324 -16 428 60 428 174C428 256 379 309 307 360L276 382C223 419 179 455 179 516C179 576 221 632 293 632C379 632 410 564 418 487L448 490C446 536 446 592 449 634Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="472" vert-adv-y="472" /></g></svg>-Iteration Scheme

We prove a new infinite-dimensional linking theorem. Using this theorem, we obtain nontrivial solutions for strongly indefinite periodic Schr&#xf6;dinger equations with sign-changing nonlinearities.

Shaowei Chen

Lishan Lin

Liqin Xiao

Nontrivial Solutions for Periodic Schr&#xf6;dinger Equations with Sign-Changing Nonlinearities

We will study the boundedness of Marcinkiewicz integrals with rough kernel on
the generalized weighted Morrey spaces. We will also prove that the commutator operators formed by a
<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3617pt" id="M1" height="12.7358pt" version="1.1" viewBox="-0.0498162 -10.3741 55.883 12.7358" width="55.883pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"/></g><g transform="matrix(.0135,0,0,-0.0135,7.871,0)"><path id="g113-78" d="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z"/></g><g transform="matrix(.0135,0,0,-0.0135,20.946,0)"><path id="g113-80" d="M699 369C699 549 575 666 407 666C186 666 23 488 23 278C23 101 145 -16 312 -16C535 -16 699 153 699 369ZM600 373C600 210 500 19 321 19C186 19 120 129 120 272C120 450 232 631 399 631C541 631 600 522 600 373Z"/></g><g transform="matrix(.0135,0,0,-0.0135,30.729,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0136,0,0,-0.0136,35.417,0)"><path id="g197-29" d="M712 0L497 285C582 312 628 378 628 466C628 609 518 662 390 662H70V0H236V271H310L505 0H712ZM491 604C551 577 584 532 584 464C584 401 550 345 491 321V604ZM447 320C428 317 411 315 392 315H236V618H397C412 618 432 613 447 610V320ZM626 44H528L363 271C392 271 422 270 451 275L626 44ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.0095,0,0,-0.0095,45.537,-5.984)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g><g transform="matrix(.0135,0,0,-0.0135,50.909,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> function and Marcinkiewicz integrals are also bounded on the generalized weighted Morrey
spaces.

Runqing Cui

Zhang Li

Boundedness of Marcinkiewicz Integrals and Their Commutators on Generalized Weighted Morrey Spaces

We consider a class of convex domains which contains non-Reinhardt domains with nonsmooth boundary. We show that the domains of this class satisfy the condition <b>Q</b>.

Jong-Do Park

Uniform Extendibility of the Bergman Kernel for Generalized Minimal Balls

Multivalued mappings and related selection theorems are fundamental tools in many branches of mathematics and applied sciences. In this paper we continue this theory and prove the existence of Caristi type selections for generalized multivalued contractions on complete metric spaces, by using some classes of functions. Also we prove fixed point and quasi-fixed point theorems.

Calogero Vetro

Francesca Vetro

Caristi Type Selections of Multivalued Mappings

For a Banach space <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M1" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 10.5029 9.04777" width="10.5029pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-89" d="M748 650H522L515 622L546 617C580 611 587 604 565 575C518 513 469 451 419 393C376 474 349 534 330 580C318 609 325 612 361 618L383 622L392 650H151L144 622C214 616 224 612 257 543L360 327C270 218 187 124 159 95C106 40 92 34 26 28L17 0H252L259 28L236 31C189 37 188 47 209 78C249 136 308 210 377 294L478 79C494 44 487 37 449 32L418 28L409 0H673L680 28C596 34 591 39 554 116L436 361C526 469 574 521 604 553C659 612 669 614 739 622L748 650Z"/></g></svg> and a measure space <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M2" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 33.3083 12.0395" width="33.3083pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.0135,0,0,-0.0135,4.688,0)"><path id="g113-153" d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z"/></g><g transform="matrix(.0135,0,0,-0.0135,14.944,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.0135,0,0,-0.0135,20.304,0)"><path id="g113-147" d="M570 168L541 175C510 88 481 71 415 71H135L378 358L167 614H348C422 614 447 608 463 588C478 571 488 546 500 500L529 504C524 559 519 633 518 650H47V628L295 324L33 23V0H531L570 168Z"/></g><g transform="matrix(.0135,0,0,-0.0135,28.406,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>, let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M3" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 48.9743 12.0395" width="48.9743pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path id="g113-78" d="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z"/></g><g transform="matrix(.0135,0,0,-0.0135,13.347,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,18.035,0)"><use xlink:href="#g113-153"/></g><g transform="matrix(.0135,0,0,-0.0135,28.291,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,33.651,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.0135,0,0,-0.0135,44.016,0)"><use xlink:href="#g113-42"/></g></svg> be the space of all <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M4" height="9.04777pt" version="1.1" viewBox="-0.0498162 -8.99795 10.5029 9.04777" width="10.5029pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-89"/></g></svg>-valued countably additive measures on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M5" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 33.3083 12.0395" width="33.3083pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,4.688,0)"><use xlink:href="#g113-153"/></g><g transform="matrix(.0135,0,0,-0.0135,14.944,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,20.304,0)"><use xlink:href="#g113-147"/></g><g transform="matrix(.0135,0,0,-0.0135,28.406,0)"><use xlink:href="#g113-42"/></g></svg> of bounded variation, with the total variation norm. In this paper we give a characterization of weakly precompact subsets of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.3616pt" id="M6" height="12.0395pt" version="1.1" viewBox="-0.0498162 -9.6779 48.9743 12.0395" width="48.9743pt"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><use xlink:href="#g113-78"/></g><g transform="matrix(.0135,0,0,-0.0135,13.347,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.0135,0,0,-0.0135,18.035,0)"><use xlink:href="#g113-153"/></g><g transform="matrix(.0135,0,0,-0.0135,28.291,0)"><use xlink:href="#g113-45"/></g><g transform="matrix(.0135,0,0,-0.0135,33.651,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.0135,0,0,-0.0135,44.016,0)"><use xlink:href="#g113-42"/></g></svg>.

Ioana Ghenciu

Weak Precompactness in the Space of Vector-Valued Measures of Bounded Variation

We give several two-weight extrapolation theorems on Lorentz spaces which extend the results of Cruz-Uribe and P&#xe9;rez (2000) and some applications for the singular integral operators, the potential type operators, and commutators.

Wenming Li

Tingting Zhang

Limei Xue

Two-Weight Extrapolation on Lorentz Spaces

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<p><em>Journal of Function Spaces</em> publishes research on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines.</p>


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Journal of Function Spaces - Table of contents 2015

title

Journal of Function Spaces publishes research on all aspects of function spaces, functional analysis, and their employment across other mathematical disciplines.

Journal of Function Spaces