Fourier Spectrum Characterizations of $$H^{p}$$ Spaces on Tubes Over Cones for $$1\le p \le \infty $$1≤p≤∞

We give Fourier spectrum characterizations of functions in the Hardy \(H^p\) spaces on tubes for \(1\le p \le \infty .\) For \(F\in L^p(\mathbb {R}^n), \) we show that F is the non-tangential boundary limit of a function in a Hardy space, \(H^{p}(T_\Gamma ),\) where \(\Gamma \) is an open cone of \(\mathbb {R}^n\) and \(T_\Gamma \) is the related tube in \(\mathbb {C}^n,\) if and only if the classical or the distributional Fourier transform of F is supported in \(\Gamma ^*,\) where \(\Gamma ^*\) is the dual cone of \(\Gamma .\) This generalizes the results of Stein and Weiss for \(p=2\) in the same context, as well as those of Qian et al. in one complex variable for \(1\le p\le \infty .\) Furthermore, we extend the Poisson and Cauchy integral representation formulas to the \(H^p\) spaces on tubes for \(p\in [1, \infty ]\) and \(p\in [1,\infty ),\) with, respectively, the two types of representations.     

L^p-gradient Estimates of Symmetric Markov Semigroups for 1 〈 p ≤ 2

For 1 〈 p ≤2, an L^p-gradient estimate for a symmetric Markov semigroup is derived in a general framework, i.e. ‖Γ^/2（Ttf）‖p≤Cp/√t‖p, where F is a carre du champ operator. As a simple application we prove that F1/2（（I- L） ^-α） is a bounded operator from L^p to L^v provided that 1 〈 p 〈 2 and 1/2〈α〈1. For any 1 〈 p 〈 2, q 〉 2 and 1/2 〈α 〈 1, there exist two positive constants cq,α,Cp,α such that ‖Df‖p≤ Cp,α‖（I - L）^αf‖p,Cq,α（I-L）^（1-α）‖Df‖q＋‖f‖q, where D is the Malliavin gradient （[2]） and L the Ornstein-Uhlenbeck operator.     

Necessary and sufficient conditions for the lipschitzian invertibility of nonlinear difference operators in the spacesl p (ℤ, ℝ) with 1≤p≤∞with 1≤p≤∞

Necessary and sufficient conditions for the Lipschitzian invertibility of the difference map

Affine Synthesis onto L p when 0<p≤1

The affine synthesis operator is shown to map the coefficient space ℓ ^p (ℤ₊×ℤ ^d ) surjectively onto L ^p (ℝ ^d ), for p∈(0,1]. Here ψ _j,k (x)=|det a _j |^1/p ψ(a _j x−k) for dilation matrices a _j that expand, and the synthesizer ψ∈L ^p (ℝ ^d ) need satisfy only mild restrictions, for example, ψ∈L ¹(ℝ ^d ) with nonzero integral or else with periodization that is real-valued, nontrivial and bounded below. An affine atomic decomposition of L ^p follows immediately:

Criterion for the Denseness of Algebraic Polynomials in the Spaces $$L_p \left( {{\mathbb{R}},d {\mu }} \right)$$ , 1 ≤ p < ∞

The criterion for the denseness of polynomials in the space established by Hamburger in 1921 is extended to the spaces , 1 p < .     

Estimates of the levy constant for\sqrt p and class number one criterion for ℚ(\sqrt p )

Let p ≡ = 3 (mod 4) be a prime, let ?( $\sqrt p $ ) be the length of the period of the expansion of $\sqrt p $ into a continued fraction, and let h(4p) be the class number of the field ?( $\sqrt p $ ). Our main result is as follows. For p > 91, h(4p)=1 if and only if ?( $\sqrt p $ ) > 0.56 $\sqrt p $ L_4p(1), where L_4p(1) is the corresponding Dirichlet series. The proof is based on studying linear relations between convergents of the expansion of $\sqrt p $ into a continued fraction. Bibliography: 13 titles.     

Über die Summe\mathop \Sigma \limits_{p \leqq N} e(\alpha p)

Ohne Zusammenfassung     

Models for CR-manifolds of type (1,K) for 3≤K≤7 and their automorphisms

Models similar to the tangent quadric are constructed for surfaces with one-dimensional complex tangent space. It is shown that these models possess the main properties of the tangent quadric. Their groups and algebras of automorphisms are found. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 452–459, March, 2000.     

Cones in the space Lp (1 ≤ p ≤ ∞) and minimax problems of mathematical statistics

The structure of cones in the spaces L_p(l p ) and their application to minimax problems of statistical assumptions are considered.Translated from Staticheskie Metody, pp. 28–30, 1978.     

The Equivalence of Hypercontractivity and Logarithmic Sobolev Inequality for q ((1 ≤ q ≤ 1) -Ornstein-Uhlenbeck Semigroup

In this paper the author proves the equivalence of hypercontractivity and logarithmic Sobolev inequality for q-Ornstein-Uhlenbeck semigroup U_t~((q))= Γ_q(e~(-t)I)(-1 ≤q ≤ 1), where Γ_q is a q-Gaussian functor.     

Toeplitz operators on hardy spaceH p (S) with 0<p≤1

LetB ⁿ be the unit ball inC ⁿ ,S is the boundary ofB ⁿ . We letL ^p (S) denote the usual Lebesgue spaces overS with respect to the normalized surface measure,H ^p (B ⁿ ) is its usua holomorphic subspace.H ^p (S) denotes the atomic Hardy spaces defined in [GL]. LetPL ² (S)H ²(B ⁿ ) denote the orthogonal projection. For eachfL (S), we useM _f L ^p (S)L ^p (S) to denote the multiplication operator, and we define the Toeplitz operatorT _f =PM _f . The paper gives a characterization theorem onf such that the Toeplitz operatorsT _f and are bounded fromH ^p (S)H ^p (B ⁿ ) with 0<p1. Also several equivalent conditions are given.     

Two-weight Lp- Lq Bounds for Positive Dyadic Operators: Unified Approach to p ≤ q and p > q

We characterize the bounded and compact generalized Volterra companion integral operators acting between the standard Fock spaces. We also obtain asymptotic estimates for the norm of these operators in terms of certain Berezin type integral transforms which involve some function theoretic properties of symbols inducing the operators. As a special case, we prove that there exist no nontrivial compact Volterra companion integral and multiplication operators on Fock spaces.     

Nondensity Property for ≤_T~h Reducibility

Let A, B are sets. We can define the reducibility ≤_T~h by A≤_T~hB iff there-exists a polynomial-bounded oracle Turing machine M such that(1) x∈A M~B accepts x.(2) There exists a polynomial q such that for all x, if M~B(x) queries B     

Hölder andL p -estimates for the\bar \partial -equation on non-smooth strictlyq-convex domainson non-smooth strictlyq-convex domains

A solution operator for the \(\bar \partial \) -equation on strictlyq-convex domains with nonsmooth boundary is constructed. It is proved that the solution satisfies optimal 1/2-Hölder andL ^p estimates.     

Multiplier and averaging operators in the Banach spaces $$\mathbf{ces(p), \, 1< p < \infty }$$ ces ( p ) , 1 < p < ∞

Positivity - The Banach sequence spaces ces(p) are generated in a specified way via the classical spaces $$ \ell _p, 1&lt; p &lt; \infty .$$ For each pair $$ 1&lt; p,q &lt; \infty...     

A Jackson Type Estimate for Shepard Operators in L p Spaces for p≧ 1

This paper considers the approximation of the Kantorovich–Shepard operators in spaces for . For the Kantorovich–Shepard operators are defined by (1.1). Then

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

New congruences for $$\ell $$-regular partitions for $$\ell \in \{5,6,7,49\}$$∈{5,6,7,49}

We find several new congruences for \(\ell \)-regular partitions for \(\ell \in \{5,6,7,49\}\) and also find alternative proofs of the congruences for 10- and 20-regular partitions which were proved earlier by Carlson and Webb (Ramanujan J 33:329–337, 2014) by using the theory of modular forms. We use certain p-dissections of \((q;q)_{\infty }\), \(\psi (q)\), \((q;q)_{\infty }^3\) and \(\psi (q^2)(q;q)_{\infty }^2\).     

ON CONDITIONAL EXPECTATION OPERATORS ON L(μ,X)(1≤p≤+∞,p≠2)

Some characterizations of the conditional expectation operators on Lebesgue-Bochnerspaces L_p(μ,X)are given,where 1≤p<∞,p≠2.Also an example is given to show thatthe characterizations of the conditional expectation operators on L_p(μ,X)are differentfrom that on L_p(μ)_z Finally,a representation of the constant-preserving contractiveprojection on spaces L_p(μ,X)is got when 0相似文献