In this paper, we show the validity of a Riesz–Thorin type interpolation theorem for linear operators acting from variable exponent Lebesgue spaces into variable exponent Morrey space in the ...framework of quasi-metric measure spaces.
The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz–Zygmund inequalities provide ...answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the q-dimensional sphere Sq, and investigate how well continuous Lp-norms of polynomials f of maximum degree n on the sphere Sq can be discretized by positively weighted Lp-sum of finitely many samples, and discuss the distortion between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points ξ1,…,ξN on Sq, the dimension q, and the degree n of the polynomials.
Let (Ω,F,P) be a complete probability space. We introduce variable Lorentz space Lp(⋅),q(Ω) defined by rearrangement functions and its related properties. Then, we establish martingale inequalities ...among these martingale Hardy-Lorentz spaces Hp(⋅),q(Ω) by applying the interpolation theorem. Furthermore, we study the boundedness of the fractional integral operator in variable martingale Hardy spaces Hp(⋅)M(Ω) and Hp(⋅),qM(Ω).
We address the optimal constants in the strong and the weak Stechkin inequalities, both in their discrete and continuous variants. These inequalities appear in the characterization of approximation ...spaces which arise from sparse approximation or have applications to interpolation theory. An elementary proof of a constant in the strong discrete Stechkin inequality given by Bennett is provided, and we improve the constants given by Levin and Stechkin and by Copson. Finally, the minimal constants in the weak discrete Stechkin inequalities and both continuous Stechkin inequalities are presented.
In this paper we prove new inequalities describing the relationship between the “size” of a function on a compact homogeneous manifold and the “size” of its Fourier coefficients. These inequalities ...can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood 17 on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in Lp(SU(2)) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on general compact homogeneous manifolds. The latter is applied to obtain conditions for the Lp-Lq boundedness of Fourier multipliers for 1<p≤2≤q<∞ on compact homogeneous manifolds as well as the Lp-Lq boundedness of general (non-invariant) operators on compact Lie groups. We also record an abstract version of the Marcinkiewicz interpolation theorem on totally ordered discrete sets, to be used in the proofs with different Plancherel measures on the unitary duals.
In a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element x we associate the eigenvalue vector λ(x) whose components are the eigenvalues of x written in the ...decreasing order. For any p∈1,∞, we define the spectral p-norm of x to be the p-norm of λ(x) in Rn. In this paper, we show that ‖x∘y‖1≤‖x‖p‖y‖q, where x∘y denotes the Jordan product of two elements x and y in V and q is the conjugate of p. For a linear transformation on V, we state and prove an interpolation theorem relative to these spectral norms. In addition, we compute/estimate the norms of Lyapunov transformations, quadratic representations, and positive transformations on V.
For θ∈(0,1)$\theta \in (0,1)$ and variable exponents p0(·),q0(·)$p_0(\cdot ),q_0(\cdot )$ and p1(·),q1(·)$p_1(\cdot ),q_1(\cdot )$ with values in 1, ∞, let the variable exponents pθ(·),qθ(·)$p_\theta ...(\cdot ),q_\theta (\cdot )$ be defined by
1/pθ(·):=(1−θ)/p0(·)+θ/p1(·),1/qθ(·):=(1−θ)/q0(·)+θ/q1(·).$$\begin{equation*} 1/p_\theta (\cdot ):=(1-\theta )/p_0(\cdot )+\theta /p_1(\cdot ), \quad 1/q_\theta (\cdot ):=(1-\theta )/q_0(\cdot )+\theta /q_1(\cdot ). \end{equation*}$$The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space Lpj(·)$L^{p_j(\cdot )}$ to the variable Lebesgue space Lqj(·)$L^{q_j(\cdot )}$ for j=0,1$j=0,1$, then
∥T∥Lpθ(·)→Lqθ(·)≤C∥T∥Lp0(·)→Lq0(·)1−θ∥T∥Lp1(·)→Lq1(·)θ,$$\begin{equation*} \Vert T\Vert _{L^{p_\theta (\cdot )}\rightarrow L^{q_\theta (\cdot )}} \le C \Vert T\Vert _{L^{p_0(\cdot )}\rightarrow L^{q_0(\cdot )}}^{1-\theta } \Vert T\Vert _{L^{p_1(\cdot )}\rightarrow L^{q_1(\cdot )}}^{\theta }, \end{equation*}$$where C is an interpolation constant independent of T. We consider two different modulars ϱmax(·)$\varrho ^{\max }(\cdot )$ and ϱsum(·)$\varrho ^{\rm sum}(\cdot )$ generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants Cmax and Csum, which imply that Cmax≤2$C_{\rm max}\le 2$ and Csum≤4$C_{\rm sum}\le 4$, as well as, lead to sufficient conditions for Cmax=1$C_{\rm max}=1$ and Csum=1$C_{\rm sum}=1$. We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that pj(·)=qj(·)$p_j(\cdot )=q_j(\cdot )$, j=0,1$j=0,1$ are Lipschitz continuous and bounded away from one and infinity (in this case, ϱmax(·)=ϱsum(·)$\varrho ^{\rm max}(\cdot )=\varrho ^{\rm sum}(\cdot )$).
-valued noncommutative symmetric spaces. By duality we may interpolate several well-known noncommutative maximal inequalities. In particular we obtain a version of Doob's maximal inequality and the ...dual Doob inequality for noncommutative symmetric spaces. We apply our results to prove the Burkholder-Davis-Gundy and Burkholder-Rosenthal inequalities for noncommutative martingales in these spaces.>
We present a uniform syntactical characterisation of the class of quasi-relevant logics which are four-valued extensions of the basic relevant logic B of Meyer and Routley. All these logics are ...obtained by the addition of suitable quasi-relevant implications to the four-valued logic of First Degree Entailment FDE. So far they were characterised axiomatically and semantically in several ways but did not obtain a special proof-theoretic treatment. To this aim a generalised form of sequent calculus called bisequent calculus (BSC) is applied. In BSC rules operate on the ordered pairs of ordinary sequents. It may be treated as the weakest kind of system in the rich family of generalised sequent calculi operating on items which are some collections of ordinary sequents, like hypersequents or nested sequents. It is shown that all logics under consideration have cut-free characterisation in BSC which satisfies the subformula property and yields decidability. It is also shown that the interpolation theorem holds for these logics if their language is enriched with additional negation.
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