Related papers: A Rigidity Phenomenon for the Hardy-Littlewood Max…

We study the frequency function (introduced by Temur) in both the discrete and continuous settings. More precisely, we extend the definition of the frequency function to the higher-dimensional continuous setting and to the uncentered…

We prove nontangential and radial maximal function characterizations for Hardy spaces associated to a non-negative self-adjoint operator satisfying Gaussian estimates on a space of homogeneous type with finite measure. This not only…

Let $f$ be a meromorphic function on the complex plane $\mathbb C$ with the maximum function of its modulus $M(r,f)$ on circles centered at zero of radius $r$. A number of classical, well-known and widely used results allow us to estimate…

nonlinearities and spatial weights of H\'enon type. Motivated by the symmetry-breaking phenomena observed in semilinear second-order problems -- such as those governed by the H\'enon equation -- we consider weighted functionals of the form…

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete…

Let $p\in(0,1]$ and $W$ be an $A_p$-matrix weight, which in scalar case is exactly a Muckenhoupt $A_1$ weight. In this article, we introduce matrix-weighted Hardy spaces $H^p_W$ via the matrix-weighted grand non-tangential maximal function…

Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $\varphi:\,\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that…

Let $\Omega\in L^1{({\mathbb S^{n-1}})}$, be a function of homogeneous of degree zero, and $M_\Omega$ be the Hardy-Littlewood maximal operator associated with $\Omega$ defined by $M_\Omega(f)(x) =…

Let $\varphi: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight. The Musielak-Orlicz Hardy space $H^{\varphi}(\mathbb…

We use the formalism of the R{\'e}nyi entropies to establish the symmetry range of extremal functions in a family of subcriti-cal Caffarelli-Kohn-Nirenberg inequalities. By extremal functions we mean functions which realize the equality…

For a continuous map $T: X\rightarrow X$ on a compact metric space $(X,d)$, we say that a function $f: X \rightarrow \mathbb{R}$ has the property $\mathscr{P}_T$ if its time averages along forward orbits of $T$ are maximized at a periodic…

We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties…

We precisely evaluate the operator norm of the uncentered Hardy-Littlewood maximal function on $L^p(\Bbb R^1)$. We also compute the operator norm of the uncentered Hardy-Littlewood maximal function over rectangles on $L^p(\Bbb R^n)$, and we…

For any operator $T$ whose bilinear form can be dominated by a sparse bilinear form, we prove that $T$ is bounded as a map from $L^1(\widetilde{M}w)$ into weak--$L^1(w)$. Our main innovation is that $\widetilde{M}$ is a maximal function…

We prove the Hardy-Littlewood theorem in two dimensions for functions whose Fourier coefficients obey general monotonicity conditions and, importantly, are not necessarily positive. The sharpness of the result is given by a counterexample,…

We prove that the Hardy--Littlewood maximal operator $M$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(X,d,\mu)$, with $1<p_-\le p_+<\infty$, over an unbounded space of homogeneous type $(X,d,\mu)$ with a Borel-semiregular measure…

We show identities of Hardy-Stein type for harmonic functions relative to integro-differential operators corresponding to general symmetric regular Dirichlet forms satisfying the absolute continuity condition. The novelty is that we…

We answer a special case of a question of T. Hytonen regarding the two weight norm inequality for the maximal function M in the affirmative, namely that there is a constant D > 1, depending only on dimension n, such that the two weight norm…

The Coifman-Fefferman inequality implies quite easily that a Calderon-Zygmund operator $T$ acts boundedly in a Banach lattice $X$ on $\mathbb R^n$ if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$. We discuss this…

Building on the author's recent work with Jan Maas and Jan van Neerven, this paper establishes the equivalence of two norms (one using a maximal function, the other a square function) used to define a Hardy space on $\R^{n}$ with the…