We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp Lp-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres...
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We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp Lp-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that Lp-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures.
Weight functions are characterized so that Hardy-Littlewood maximal operator is bounded in certain spaces. The reverse weak type estimates with applications to some singular integrals and to the class L(1 + log(+) L) ...
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Weight functions are characterized so that Hardy-Littlewood maximal operator is bounded in certain spaces. The reverse weak type estimates with applications to some singular integrals and to the class L(1 + log(+) L) of Zygmund are established. These results are also compared with the ones in Euclidean case which are obtained by K.F. Andersen and W.S. Young, thereby showing the differences between the two cases. We introduce a weak type estimate for a new class of maximal function and employ it to deduce a special result on singular operators over a local field which is obtained by K. Phillips and M. Taibleson.. (C) 2009 Elsevier Inc. All rights reserved.
Our aim in this paper is to deal with Sobolev embeddings for Riesz potentials of variable order with functions in variable exponent Musielak-Orlicz-Morrey spaces.
Our aim in this paper is to deal with Sobolev embeddings for Riesz potentials of variable order with functions in variable exponent Musielak-Orlicz-Morrey spaces.
Let E be a UMD Banach space, and L a positive self-adjoint operator in L-2 of Laplace type, for which the imaginary powers L-it form a C-0-group of exponential growth 0 less than or equal to alpha 0 and phi > alph...
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Let E be a UMD Banach space, and L a positive self-adjoint operator in L-2 of Laplace type, for which the imaginary powers L-it form a C-0-group of exponential growth 0 less than or equal to alpha < π on L-p(E), where 1 < p < ∞. Suppose G(z) is holomorphic inside and on the boundary of the sector {z : z ¬equal;0, |arg z| ≤ φ}, and z(κ)G(z) → 0 uniformly as z → ∞ for some κ > 0 and phi > alpha. Then G(tL) (t > 0) defines a bounded family of linear operators on L-p(E);and the maximal operator f --> sup(t>0) parallel toG(tL)fparallel to(E) is bounded on the domain of log L. The proof uses transference methods. These hypotheses hold for the maximal solution operators for the heat, wave and Schrodinger equations, and for Cesaro sums.
Given a smooth, compactly supported hypersurface S in Rn that does not pass through the origin, and denoting by tS the surface dilated by a factor t>0, we can consider the averaging operator defined for functions f...
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Given a smooth, compactly supported hypersurface S in Rn that does not pass through the origin, and denoting by tS the surface dilated by a factor t>0, we can consider the averaging operator defined for functions f-5, the Schwartz class of functions, byformula herewhere dà is Lebesgue measure on S. We can now define a maximal average operator,formula hereIn the case where S is Sn-1, the sphere of unit radius in Rn, we are looking at Stein's spherical maximal function, as treated in his paper [10]. Stein proved that M is a bounded operator on the Lp spaces if and only if p>n/(n-1) when n>3. Subsequently, Bourgain [2] showed that if S is any compactly supported smooth curve with non-vanishing Gaussian curvature, then M will be bounded on Lp, if and only if p>2, thus dealing with the case of the circular maximal function in the plane. (For related results, see also [6] and [7]). In the case where S is a curve whose curvature vanishes to order at most m-2 at a single point, Iosevich [4] showed that M is bounded on Lp for p>m, and unbounded if p = m. If we study curves given by “(s) = (s, ³(s)+1), s-[0, 1], for some suitably smooth ³, where ³(0) = ³2(0) = & = ³(m-1)(0) ` ³(m)(0)>0, then we can reinterpret his results as follows. Defineformula herefor Schwartz functions f. Iosevich proved thatformula herefor p>2. If we note that º(s), the curvature of the curve “(s) is approximately 2-k(m-2) whenever s-[2-k, 21-k], then we have that the operatorformula hereis bounded on Lp for some p>2, if à is sufficiently large, sinceformula herewhich is finite so long as Ã>(m/p1) (m-2)-1. If we want to choose à independent of m>2, the type of the curve, such that Mà is bounded on Lp for some fixed p>2, then clearly we can take à = 1/p. In this paper we show that Mà will be bounded on Lp for p>max{Ã-1, s} for a class of infinitely flat, convex curves in the plane. Counterexamples will show that this is the best possible result, in that there exist flat curves for which Mà is unbounded f
In this paper we study certain maximal functions for a class of Chebli-Trimeche hypergroups. Versions of the standard maximal theorems for these maximal functions are established and some applications are given.
In this paper we study certain maximal functions for a class of Chebli-Trimeche hypergroups. Versions of the standard maximal theorems for these maximal functions are established and some applications are given.
The author establishes the L-p boundedness for a class of maximal functions related to singular integrals associated to surfaces of revolution on product domains with tough kernels in L(log L)(Sn-1). (c) 2008 Elsevier...
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The author establishes the L-p boundedness for a class of maximal functions related to singular integrals associated to surfaces of revolution on product domains with tough kernels in L(log L)(Sn-1). (c) 2008 Elsevier Inc. All rights reserved.
We consider convolution operators on R-n of the form T(P)f(x) = integral(Rm) f(x - P(y))K(y)dy, where P is a polynomial defined on R-m with values in R-n and K is a smooth Calderon-Zygmund kernel on R-m. A maximal ope...
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We consider convolution operators on R-n of the form T(P)f(x) = integral(Rm) f(x - P(y))K(y)dy, where P is a polynomial defined on R-m with values in R-n and K is a smooth Calderon-Zygmund kernel on R-m. A maximal operator M-P can be constructed in a similar fashion. We discuss weak-type 1-1 estimates for T-P and M-P and the uniformity of such estimates with respect to P. We also obtain L-p-estimates for "supermaximar' operators, defined by taking suprema over P ranging in certain classes of polynomials of bounded degree.
In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov *** boundedness and continuity o...
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In this paper,we systematically study several classes of maximal singular integrals and maximal functions with rough kernels in Fβ(S^n-1),a topic that relates to the Grafakos-Stefanov *** boundedness and continuity of these operators on Triebel-Lizorkin spaces and Besov spaces are discussed.
Abstract In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a conseque...
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Abstract In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space L log L（Sn-1）. Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.
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