Related papers: The Geometry of the Dyadic Maximal Operator
We prove sharp $L^1$ inequalities for the dyadic maximal function $M_T\phi$ when $\phi$ satisfies certain $L^1$ and $L^{\infty}$ conditions
With rectangular doubling weight, a~generalized Hardy-Littlewood-Sobolev inequality for rectangular fractional integral operators is verified. The result is a~nice application of $M$-linear embedding theorem for dyadic rectangles.
We provide an alternative proof and expression of the Bellman function of the dyadic maximal operator in connection with the Dyadic Carleson Imbedding Theorem, which appears in [10]. We also evaluate the Bellman function of four variables…
Given a space of homogeneous type we give sufficient conditions on a variable exponent {p(.)} so that the fractional maximal operator {M_{\eta}} maps {L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = {\eta}}. In the endpoint case we…
We introduce a geometric approach of integral curves for functional inequalities involving directional derivatives in the general context of differentiable manifolds that are equipped with a volume form. We focus on Hardy-type inequalities…
We study different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy-Littlewood maximal averaging operator. In particular, we analyze the connections between the doubling condition, having finite…
Integration over curved manifolds with higher codimension and, separately, discrete variants of continuous operators, have been two important, yet separate themes in harmonic analysis, discrete geometry and analytic number theory research.…
We establish an analog Hardy inequality with sharp constant involving exponential weight function. The special case of this inequality (for n=2) leads to a direct proof of Onofri inequality on S^2.
A refinement of the Hardy inequality has been presented by use of superquadratic function.
We obtain sharp Hardy inequalities on antisymmetric functions where antisymmetry is understood for multi-dimensional particles. Partially it is an extension of the previously published paper \cite{HL}, where Hardy's inequalities were…
The Fefferman-Stein type inequalities for strong maximal operator and directional maximal operator are verified with composition of the Hardy-Littlewood maximal operator in the plane.
In this paper the complete solution of the restricted inequalities for supremal operators are given. The boundedness of the composition of supremal operators with the Hardy and Copson operators in weighted Lebesgue spaces are characterized.
In this paper we present Hardy type inequalities for magnetic Dirichlet forms with singular integral weights. We analyze the local and global optimality of the integral weight and discuss several examples in details. An application of our…
We get the sharp bound for weak type $(1,1)$ inequality for $n$-dimensional Hardy operator. Moreover, the precise norms of generalized Hardy operators on the type of Campanato spaces are obtained. As applications, the corresponding norms of…
We study generalized Poincar\'e inequalities. We prove that if a function satisfies a suitable inequality of Poincar\'e type, then the Hardy-Littlewood maximal function also obeys a meaningful estimate of similar form. As a by-product, we…
In this paper, we are devoted to studying some sharp bounds for Hardy type operators on mixed radial-angular type function spaces. In addition, we will establish the sharp weak-type estimates for the fractional Hardy operator and its…
We deal with Dirac operators with external homogeneous magnetic fields. Hardy-type inequalities related to these operators are investigated: for a suitable class of transversal magnetic fields, we prove a Hardy inequality with the same best…
This paper focuses on the operator norm of the truncated Hardy-Littlewood maximal operator $M^b_a$ and the strong truncated Hardy-Littlewood maximal operator $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$, respectively. We first present the…
We find best constants in several dilation invariant integral inequalities involving derivatives of functions. Some of these inequalities are new and some were known without best constants. The contents: 1. Estimate for a quadratic form of…
To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is…