We study approximation properties of linear sampling operators in the spaces $L_p$ for $1\le p<\infty$. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of a function in $L_p$ and discrete information on the behaviour of a function at sampling points. The new measure of smoothness enables us to improve and extend several classical results of approximation theory to the case of linear sampling operators. In particular, we obtain matching direct and inverse approximation inequalities for sampling operators in $L_p$, find the exact order of decay of the corresponding $L_p$-errors for particular classes of functions, and introduce a special $K$-functional and its realization suitable for studying smoothness properties of sampling operators.
We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a by-product we derive new results about multi-window Gabor frames with respect to vectors of Hermite functions or totally positive functions.
In this article, we establish sharp weak endpoint estimates for $p$-adic fractional Hardy operator. In addition, sharp weak bounds for p-adic Hardy operator on p-adic central Morrey space are also obtained.
We prove the sharp weighted-$L^2$ bounds for the strong-sparse operators introduced in \cite{KaragulyanM}. The main contribution of the paper is the construction of a weight that is a lacunary mixture of dual power weights. This weights helps to prove the sharpness of the trivial upper bound of the operator norm.
We obtain sharp $L^p$ bounds for oscillatory integral operators with generic homogeneous polynomial phases in several variables. The phases considered in this paper satisfy the rank one condition which is an important notion introduced by Greenleaf, Pramanik and Tang. Under certain additional assumptions, we can establish sharp damping estimates with critical exponents to prove endpoint $L^p$ estimates.
We use broad-narrow method to estabish the sharp $L^4$ decay estimate for a class of degenerate oscillatory integral operators in $(2+1)$ dimensions. Especially, the model phase function is \[xt^2+y^2t,\] a cubic homogeneous polynomial which is degenerate in the sense of [28].
In this article, we address pointwise sparse domination for multilinear Calderón-Zygmund operators on upper doubling, geometrically doubling metric measure spaces. As a consequence, we have obtained sharp quantitative weighted estimates for multilinear Calderón-Zygmund operators.
Let $L f(x):=-\frac{d^2}{dx^2}f(x)-\frac{ r}{x}\frac{d}{dx}f(x),\quad x>0, r>0$ be the Bessel operator on $((0,\infty), |\cdot|, x^rdx)$. In this paper, we prove the sharp weak type $(1,1)$ estimate for the imaginary power $L^{iα}, α\in \mathbb R$, of the Bessel operator.
The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space $%H_{p}$ to the Lebesgue space $L_{p}$ for all $0<p\leq 1.$ We also prove that the result is sharp in a particular sense.
The sharp range of $L^p$-estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments.
The sharp range of $L^p$-estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a general signature assumption on the phase. This simultaneously generalises earlier work of the authors and Guth, which treats the maximal signature case, and also work of Stein and Bourgain--Guth, which treats the minimal signature case.
In this paper we prove the sharp boundedness for a fractional type operator given by a kernel that satisfy a $L^{α,r'}$-Hörmander conditions and a fractional size condition, where $0<α<n$ and $1< r'\leq \infty$. To prove this result we use a new appropriate sparse domination which we provide in this work. For the case $r'=\infty$ we recover the sharp boundedness for the fractional integral, $I_α$, proved in [Lacey, M. T., Moen, K., Pérez, C., Torres, R. H. (2010). Sharp weighted bounds for fractional integral operators. Journal of Functional Analysis, 259(5), 1073-1097.]
In this paper, we consider the degenerate and singular oscillatory integral operator with a singular kernel which is not a Calderón-Zygmund kernel and satisfies suitable size and derivative conditions related to a real parameter $μ$. For any given homogeneous polynomial phases, except monomial phases, of degreee $n$, we give the range of $p$ for which the sharp decay rate $-\frac{1-μ}{n}$ on $L^2$ spaces can be preserved on $L^p$ spaces.
In this paper, Hardy type operator $H_β$ on $\bR^{n}$ and its adjoint operator $H_β^{*}$ are investigated. We use novel methods to obtain two main results. One is that we obtain the operators $H_β$ and $H_β^{*}$ being bounded from $L^{p}(|x|^α)$ to $L^{q,\infty}(|x|^γ)$, and the bounds of the operators $H_β$ and $H_β^{*}$ are sharp worked out. In particular, when $α=γ=0$, the norm of $H_β$ is equal to $1$. The other is that we study limiting weak-type behavior for the operator $H_β$ and its optimal form was obtained.
In this paper, by using the rotation method, we calculate that the sharp bound for $n$-dimensional Hardy operator $\mathcal{H}$ on mixed radial-angular spaces. Furthermore, we also obtain the sharp bound for $n$-dimensional fractional Hardy operator $\mathcal{H}_β$ from $L^p_{|x|}L_θ^{\bar{p}}({\Bbb R}^n)$ to $L^q_{|x|}L_θ^{\bar{q}}({\Bbb R}^n)$, where $0<β<n$, $1<p,q,\bar{p},\bar{q}<\infty$ and $1/p-1/q=β/n$. By using duality, the corresponding results for the dual operators $\mathcal{H}^*$ and $\mathcal{H}^*_β$ are also established. In addition, the sharp weak-type estimate for $\mathcal{H}$…
We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard $α$-fractional singular integrals, fractional integral operators, Marcinkiewicz integral operators, and square functions. These bounds are knows to be sharp in many cases, and our main new result is the optimal bound $$[w]_{A_{p,q}}^{\frac{1}{q}}[w^{q}]_{A_{\infty}}^{\frac{1}{2}-\frac{1}{p}}\lesssim[w]_{A_{p,q}}^{\frac{1}{2}-\fracα{d}}$$ for proper conditions which satisfy that three index $p$, $q$ and $α$ ensure weak type norm of fractional square functions on $L^{q}(w^{q})$ with $p…
Given $1\leq q<p<\infty$ quantitative weighted L^p estimates, in terms of Aq weights, for vector valued maximal functions, Calderón-Zygmund operators, commutators and maximal rough singular integrals are obtained. The results for singular operators will rely upon suitable convex body domination results, which in the case of commutators will be provided in this work, obtaining as a byproduct a new proof for the scalar case as well.
For a class of sparse operators including majorants of singular integral, square function, and fractional integral operators in a uniform manner, we prove off-diagonal two-weight estimates of mixed type in the two-weight and $A_{\infty}$-characteristics. These bounds are known to be sharp in many cases; as a new result, we prove their sharpness for fractional square functions.
Let $\mathcal{B}$ denote a nonempty translation invariant collection of intervals in $\mathbb{R}^n$ (which we regard as a rare basis), and define the associated geometric maximal operator $M_\mathcal{B}$ by $$M_\mathcal{B}f(x) = \sup_{x \in R \in \mathcal{B}} \frac{1}{|R|}\int_R |f|.$$ We provide a sufficient condition on $\mathcal{B}$ so that the estimate $$ |\{x \in \mathbb{R}^n : M_{\mathcal{B}}f(x) > α\}|\leq C_n \int_{\mathbb{R}^{n}} \frac{|f|}α\left(1+\log^+\frac{|f|}α\right)^{n-1} $$ is sharp. As a corollary we obtain sharp weak type estimates for maximal operators associated to several…
