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.
We prove that the maximal functions associated with a Zygmund dilation dyadic structure in three-dimensional Euclidean space, and with the flag dyadic structure in two-dimensional Euclidean space, cannot be bounded by...
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We prove that the maximal functions associated with a Zygmund dilation dyadic structure in three-dimensional Euclidean space, and with the flag dyadic structure in two-dimensional Euclidean space, cannot be bounded by multiparameter sparse operators associated with the corresponding dyadic grid. We also obtain supplementary results about the absence of sparse domination for the strong dyadic maximal function.
Let G be a connected and finite graph with the set of vertices V and the set of edges E. Let MG be the Hardy-Littlewood maximal function defined on graph G and M alpha,G(0 BVp(G) is bounded and M alpha,G:lp(V)-> B...
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Let G be a connected and finite graph with the set of vertices V and the set of edges E. Let MG be the Hardy-Littlewood maximal function defined on graph G and M alpha,G(0 <=alpha<1) be its fractional version. In this paper, the regularity problems related to MG and M alpha,G will be studied. We show that MG:BVp(G)-> BVp(G) is bounded and M alpha,G:lp(V)-> BVq(G) is bounded and continuous for all 0
functions
of bounded p-variation on V. The operator norms of MG and M alpha,G have also been investigated.
Let M be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup M(Delta)f (x) : = sup(t>0) vertical bar e (t Delta)f(x)...
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Let M be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup M(Delta)f (x) : = sup(t>0) vertical bar e (t Delta)f(x)vertical bar where Delta is the Laplace-Beltrami operator acting on M. Then, by combining the subordination formula with the previous result, we obtain the weak type (1;1) and L-p boundedness of the maximal function M(root L)(k)f(x) : = sup(t>0) vertical bar(t root L)(k)e(-t) (root L) f (x)vertical bar on L-p( M) for 1 < p <= infinity where k is a nonnegative integer and L is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of M and the smoothness of the operators' kernels.
Besicovitch proved that if f is an integrable function on R2 whose associated strong maximal function MSf is finite a.e., then the integral off is strongly differentiable. On the other hand, Papoulis proved the existe...
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Besicovitch proved that if f is an integrable function on R2 whose associated strong maximal function MSf is finite a.e., then the integral off is strongly differentiable. On the other hand, Papoulis proved the existence of an integrable function on R2 (taking on both positive and negative values) whose integral is strongly differentiable but whose associated strong maximal function is infinite on a set of positive measure. In this paper, we prove that if n >= 2 and if f is a measurable nonnegative function on Rn whose integral is strongly differentiable and moreover such that f (1 +log+ f )n-2 is integrable, then MSf is finite a.e. We also show this result is sharp by proving that, if phi is a continuous increasing function on [0, infinity) such that phi(0) = 0 and with phi(u) = o(u(1 + log+ u)n-2) (u -> infinity), then there exists a nonnegative measurable function f on Rn such that phi(f) is integrable on Rn and the integral off is strongly differentiable, although MSf is infinite almost everywhere.(c) 2023 Elsevier Inc. All rights reserved.
We prove the sharp mixed A(p) - A(infinity) weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely ||M||(Lp,q(w)) less than or similar to(p,q,n) [w](Ap)(1/p) [sig...
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We prove the sharp mixed A(p) - A(infinity) weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely ||M||(Lp,q(w)) less than or similar to(p,q,n) [w](Ap)(1/p) [sigma](A infinity)(1/min(p,q)), where sigma = w(1/1-p). Our method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for M in a dual setting. (c) 2022 Elsevier Inc. All rights reserved.
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.
We provide quantitative weighted estimates for theLp(w) norm of a maximal operator associated to cube skeletons in Rn. The method of proof differs from the usual in the area of weighted inequalities since there are no...
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We provide quantitative weighted estimates for theLp(w) norm of a maximal operator associated to cube skeletons in Rn. The method of proof differs from the usual in the area of weighted inequalities since there are no covering arguments suitable for the geometry of skeletons. We use instead a combinatorial strategy that allows to obtain, after a linearization and discretization, Lp bounds for the maximal operator from an estimate related to intersections between skeletons and k-planes.
Inspired by a question of Lie, we study boundedness in subspaces of L-1(R) of oscillatory maximal functions. In particular, we construct functions in L-1(R) which are never integrable under action of our class of maxi...
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Inspired by a question of Lie, we study boundedness in subspaces of L-1(R) of oscillatory maximal functions. In particular, we construct functions in L-1(R) which are never integrable under action of our class of maximal functions. On the other hand, we prove that these maximal functions map certain classes of spaces resembling Sobolev spaces into L-1(R) continuously under mild curvature assumptions on the phase gamma. (C) 2020 Elsevier Inc. All rights reserved.
Let M-(u), H-(u) be the maximal operator and Hilbert transform along the parabola (t, ut(2)). For U subset of (0, infinity) we consider L-p estimates for the maximal functions sup (u is an element of U)|M((u))f| and s...
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Let M-(u), H-(u) be the maximal operator and Hilbert transform along the parabola (t, ut(2)). For U subset of (0, infinity) we consider L-p estimates for the maximal functions sup (u is an element of U)|M((u))f| and sup (u is an element of U)|H((u))f|, when 1 < p <= 2. The parabolas can be replaced by more general non-flat homogeneous curves.
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