We first extend Calderón's transfer principle to weighted spaces in various different settings under suitable assumptions. Then we apply our results for some inequalities on the real line obtained by the author to prove corresponding inequalities in ergodic theory and ergodic $H^1$ spaces as well.
Variable Muckenhoupt weights are considered in variable exponent Lebesgue spaces. Applications are given for polynomial approximation in these spaces. Boundedness of averaging operator is proved to gain a transference result. Almost all weighted norm inequalities are obtained using this transference result. Potential type approximate identities are considered. Translations of Steklov operator are proved to be bounded in these spaces. K-functional is a good apparatus for measuring smoothness properties of functions given in these function classes. Main inequalities of approximation are derived.
In this paper, we establish a noncommutative analogue of Calderón's transference principle, which allows us to deduce noncommutative ergodic maximal inequalities from the special case---operator-valued maximal inequalities. As applications, we deduce dimension-free estimates of noncommutative Wiener's maximal ergodic inequality and noncommutative Stein-Calderón's maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener's pointwise ergodic theorem, we construct a dense subset on which pointwise convergence holds following a som…
We extend the results of Jones, Rosenblatt, and Wierdl concerning higher-dimensional oscillation in ergodic theory in a variety of ways. We do so by transference to the integer lattice, where we employ technique from (discrete) harmonic analysis.
We provide a version of the transference principle. It says that certain optimization problems for functions on the circle, the interval, and the line have the same answers. In particular, we show that the sharp constants in the John--Nirenberg inequalities for naturally defined $\mathrm{BMO}$-spaces on the circle, the interval, and the line coincide. The same principle holds true for the Reverse Hölder inequality for Muckenhoupt weights.
In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of oscillation estimates in both the single and several parameter settings. We establish a number of new oscillation inequalities and give new proofs for known results with elementary arguments.
It is a well known general principle that the Fourier transform of a random measure is small, except at the zero frequency, in various senses for appropriate notions of randomness. In this note we develop analogues of this principle for two classes of random multilinear operators. Two ergodic theoretic applications, involving correlations over randomly generated sparse subsequences, are obtained as corollaries of the main results.
In this paper we establish uniform oscillation estimates on $L^p(X)$ with $p\in(1,\infty)$ for the polynomial ergodic averages. This result contributes to a certain problem about uniform oscillation bounds for ergodic averages formulated by Rosenblatt and Wierdl in the early 1990's. We also give a slightly different proof of the uniform oscillation inequality of Jones, Kaufman, Rosenblatt and Wierdl for bounded martingales. Finally, we show that oscillations, in contrast to jump inequalities, cannot be seen as an endpoint for $r$-variation inequalities.
We extend the main result of Harrow, Kolla, and Schulman -- the existence of dimension-free $L^2$-bounds for the spherical maximal function in the hypercube -- to all $L^p, p > 1$. Our approach is motivated by the spectral technique developed by Stein and Nevo and Stein in the context of pointwise ergodic theorems on general groups. We provide an example which demonstrates that no dimension-free weak-type (1-1) bound exists at the endpoint.
In this paper, we establish $\mathcal B$-valued variational inequalities for differential operators, ergodic averages and symmetric diffusion semigroups under the condition that Banach space $\mathcal B$ has martingale cotype property. These results generalize, on the one hand Pisier and Xu's result on the variational inequalities for $\mathcal B$-valued martingales, on the other hand many classical variational inequalities in harmonic analysis and ergodic theory. Moreover, we show that Rademacher cotype $q$ is necessary for the $\mathcal B$-valued $q$-variational inequalities. As applications…
In this paper, we establish UMD lattice-valued variational inequalities for differential operators, ergodic averages and analytic semigroups. These results generalize, on the one hand some scalar-valued variational inequalities in ergodic theory, on the other hand Xu's very recent result on UMD lattice-valued maximal inequality. As a consequence, we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence.
Various topics in stochastic processes have been considered in the abstract setting of Riesz spaces, for example martingales, martingale convergence, ergodic theory, AMARTS, Markov processes and mixingales. Here we continue the relaxation of conditional independence begun in the study of mixingales and study mixing processes. The two mixing coefficients which will be considered are the $α$ (strong) and $φ$ (uniform) mixing coefficients. We conclude with mixing inequalities for these types of processes. In order to facilitate this development, the study of generalized $L^1$ and $L^\infty$ space…
Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions of finite degree (IFFD) in some variable exponent Lebesgue space of real functions defined on $\boldsymbol{R}:=\left( -\infty ,+\infty \right) $. To do this we employ a transference theorem which produce norm inequalities starting from norm inequalities in $\mathcal{C}(\boldsymbol{R})$, the class of bounded uniformly continuous functions defined on $\boldsymbol{R}$. Let $B\subseteq \boldsymbol{R}$ be a measurable set, $p\left( x\right) :B\rightarrow \lbrack 1,\infty )$ be a me…
