Strongly Singular Convolution Operators on Herz-Type Hardy Spaces with Variable Exponent
We investigate the boundedness of the strongly singular convolution operators on Herz-type Hardy spaces with variable exponent.
The theory of function spaces with variable exponents has been extensively studied by researchers since the work of Kováik and Rákosník  appeared in 1991. In [2, 3] the authors defined the Herz-type Hardy spaces with variable exponent and gave some characterizations for them. In [4–7], the authors proved the boundedness of some integral operators on variable function spaces.
Given an open set and a measurable function , denotes the set of measurable functions defined on such thatholds for some .
The set is a Banach function space when it is equipped with the Luxemburg-Nakano norm as follows: The space is regarded as the variable space, since it generalized the standard space: if is constant, then is isometrically isomorphic to .
The space is defined by Define to be the set of such thatDefine to be the set of such thatandDenote
Let . The Hardy-Littlewood maximal operator is defined bywhere . Let be the set of such that the Hardy-Littlewood maximal operator is bounded on .
Lemma 1 (see ). If and satisfiesand then , that is, the Hardy-Littlewood maximal operator is bounded on .
In addition, we denote the Lebesgue measure and the characteristic function of a measurable set by and , respectively. The notation means that there exist two constants such that .
Next we recall the definition of the Herz spaces with variable exponent. Let and for . Denote and as the sets of all positive and nonnegative integers, respectively, for , if , and .
Definition 2 (see ). Let , and . The homogeneous Herz space with variable exponent is defined bywhereThe nonhomogeneous Herz space with variable exponent is defined by whereIn , the authors gave the definition of the Herz-type Hardy space with variable exponent and the atomic decomposition characterizations. denotes the space of Schwartz functions, and denotes the dual space of . Let be the grand maximal function of defined bywhereand ; is the nontangential maximal operator defined bywith .
Definition 3 (see ). Let , and .
(i) The homogeneous Herz-type Hardy space with variable exponent is defined byand(ii) The nonhomogeneous Herz-type Hardy space with variable exponent is defined byand
For , we denote by the largest integer less than or equal to . is the same as in Lemma 9.
Definition 4 (see ). Let , and nonnegative integer .(i)A function on is said to be a central -atom, if it satisfies(1)supp (2)(3)(ii)A function on is said to be a central -atom of restricted type, if it satisfies conditions (2), (3) and(1′)supp
If for some in Definition 4, then the corresponding central -atom is called a dyadic central -atom.
Lemma 5 (see ). Let and . Then (or ) if and only ifwhere each is a central -atom (or central -atom of restricted type) with support contained in and (or ). Moreover,where the infimum is taken over all above decomposition of .
Let be a smooth radial cut-off function such that if and if . Define the multiplierswhere . The kernel for is very singular. Roughly speaking, it looks likewhere . Indeed the cancellation is minimal and if one makes a quick computation for , we haveThe study of these operators in the context of spaces was carried out by Hirschman  and Wainger . Sharp endpoint estimates were obtained by Fefferman and Stein in  via the duality of and BMO. Weighted norm and weak(1,1) estimates were established by Chanillo in . The boundedness of these operators on the weighted Herz-type Hardy spaces was proved by Xiaochun Li and Shanzhen Lu in .
Motivated by [2, 14], we will study the boundedness of the strongly singular convolution operators on Herz-type Hardy spaces with variable exponent. The main results are as follows.
2. Preliminary Lemmas
Referring to the variable space, there are some important lemmas as follows.
Lemma 8 (see ). Let . If and , then is integrable on andwhere The above inequality is named generalized Hölder’s inequality with respect to the variable space.
Lemma 9 (see ). Let . Then there exists a positive constant such that, for all balls in and all measurable subsets ,andhold, where and are constants with .
Throughout this paper is the same as in Lemma 9.
Lemma 10 (see ). Suppose . Then there exists a constant such that, for all balls in ,
Lemma 11 (see ). Define a variable exponent by for Then we havefor all measurable functions and .
Lemma 12 (see ). Let satisfy conditions (8) and (9) in Lemma 1. Thenfor every cube (or ball) , where .
A nonnegative locally integrable function on is said to belong to , if there is a constant such that where ; denotes a cube in with its sides parallel to the coordinate axes.
The weighted boundedness of has been proved by Chanillo .
Lemma 13 (see ). Let . Then
Lemma 14 (see ). Given a family and an open set , assume that for some and for every ,Given such that satisfies (8) and (9) in Lemma 1, then for all such that Since , by Lemmas 13 and 14 it is easy to get the -boundedness of the strongly singular convolution operators .
To prove our main results, we also need the following lemmas.
Lemma 15 (see ). The kernel for the multiplier operator is given bywith . Here and depend only on .
Lemma 16 (see ). Let and . Then
3. The Proof of Main Results
Firstly we give the proof of Theorem 6.
Proof of Theorem 6. Let . By Lemma 5, we havewherethe infimum is taken over the above decomposition of , and is a dyadic central -atom with the support . Then we haveWe first estimate ; by and the -boundedness of we haveNow we estimate . LetBy Lemma 15 and the Minkowski inequality, we have To estimate the term , we need the pointwise estimate for .
Let . By generalized Hölder’s inequality we have Therefore, by , , Lemmas 9 and 10, the Minkowski inequality, and generalized Hölder’s inequality we haveWhat remains is estimating . Let for some , where is the same as the above. Then it follows thatTo estimate the term , we need the pointwise estimate for . Let . Then, by the vanishing moment condition on , we haveFrom the condition of , , if , it follows thatNote that ; that is, . Since , , and , then by Lemmas 9 and 10 we haveNow to estimate , we split as follows: where is the same as in Lemma 16 and let satisfy .
Applying the mean value theorem to the term brackets in the integrand of , then for we have the pointwise estimate for as follows:On the other hand, since , by the Minkowski inequality we getFor , using , the pointwise estimate for , and Lemmas 9 and 10 we haveFinally, we estimate . Noting that , we get Noting , we denote and .
When and , by Lemma 12 we haveWhen we haveSo we obtainIn similar method we can obtainandThus by Lemmas 11, 12, and 16 we have