arXiv subjects
Publications and source records attributed to Sakin Demir.
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.
Let $\mathcal{H}$ be a complex Hilbert space and $T:\mathcal{H}\to \mathcal{H}$ be a contraction. Let $$A_nf=\frac{1}{n}\sum_{j=1}^nT^jf$$ for $f\in \mathcal{H}$. Let $(n_k)$ be a lacunary sequence, then there exists a constant $C_1>0$ such that $$\sum_{k=1}^\infty\|A_{n_{k+1}}f-A_{n_k}f\|_{\mathcal{H}}\leq C_1\|f\|_{\mathcal{H}}$$ for all $f\in \mathcal{H}$.\\ \indent Let $(n_k)$ be a lacunary sequence, and let $\mathbb{N}$ be the set of natural numbers. Then there exists a constant $C_2>0$ such that $$\sum_{k=1}^\infty\sup_{\substack{n_k\leq m< n_{k+1}\\m\in \mathbb{N}}}\|A_m(T)f-A_{n_k}(T)f\|_{\mathcal{H}}\leq C_2\|f\|_{\mathcal{H}}$$ for all $f\in \mathcal{H}$.
In this research we introduce the Banach space valued $H^p$ spaces with $A_p$ weight, and prove the following results: Let $\mathbb{A}$ and $\mathbb{B}$ Banach spaces, and $T$ be a convolution operator mapping $\mathbb{A}$-valued functions into $\mathbb{B}$-valued functions, i.e., $$Tf(x)=\int_{\mathbb{R}^n}K(x-y)\cdot f(y)\, dy,$$ where $K$ is a strongly measurable function defined on $\mathbb{R}^n$ such that $\|K(x)\|_{\mathbb{B}}$ is locally integrable away from the origin. Suppose that $w$ is a positive weight function defined on $\mathbb{R}^n$, and that i) For some $q\in [1, \infty ]$, there exists a positive constant $C_1$ such that $$\int_{\mathbb{R}^n}\|Tf(x)\|^q_{\mathbb{B}}w(x)\, dx\leq C_1\int_{\mathbb{R}^n}\|f(x)\|_{\mathbb{A}}^q w(x)\,dx$$ for all $f\in L^q_{\mathbb{A}}(\mathbb{R}^n)$. ii) There exists a positive constant $C_2$ independent of $y\in\mathbb{R}^n$ such that $$\int_{|x|>2|y|}\|K(x-y)-K(x)\|_{\mathbb{B}}\, dx<C_2.$$ Then there exists a positive constant $C_3$ such that $$\|Tf\|_{L^1_{\mathbb{B}}(w)}\leq C_3\|f\|_{H^1_{\mathbb{A}}(w)}$$ for all $f\in H^1_{\mathbb{A}}(w)$. Let $w\in A_1$. Assume that $K\in L_{\rm{loc}}(\mathbb{R}^n\backslash \{0\})$ satisfies $$\|K\ast f\|_{L^2_{\mathbb{B}}(w)}\leq C_1\|f\|_{L^2_{\mathbb{A}}(w)}$$ and $$\int_{|x|\geq C_2|y|}\|K(x-y)-K(x)\|_{\mathbb{B}}w(x+h)\, dx\leq C_3w(y+h)\;\;\;(\forall y\neq 0, \forall h\in\mathbb{R}^n) $$ for certain absolute constants $C_1$, $C_2$, and $C_3$. Then there exists a positive constant $C$ independent of $f$ such that $$\|K\ast f\|_{L^1_{\mathbb{B}}(w)}\leq C\|f\|_{H^1_{\mathbb{A}}(w)}$$ for all $f\in H^1_{\mathbb{A}}(w)$.
Let $ϕ\in \mathscr{S}$ with $\intϕ(x)\, dx=1$, and define $$ϕ_t(x)=\frac{1}{t^n}ϕ(\frac{x}{t}),$$ and denote the function family $\{ϕ_t\ast f(x)\}_{t>0}$ by $Φ\ast f(x)$. Suppose that there exists a constant $C_1$ such that $$\sum_{t>0} |\hatϕ_t(x)|^2 0$ such that $$\|\mathscr{V}_2(Φ\ast f)\|_{L^p}\leq C_2\|f\|_{H^p},\;\;\frac{n}{n+1} 0$ for some constant $C_3>0$.
Suppose that $\{a_j\}\in \ell^1$, and suppose that for any sequence $(t_n)$ of integers there exits a constant $C_1>0$ such that $$\sharp\left\{k\in\mathbb{Z}:\sup_{n\geq 1}\left|\sum_{i\in \mathcal{B}_n-t_n} \!\!\!\raise{1.9ex}\hbox{$\scriptsize\prime$}\; \frac{a_{k+i}}{i}\right|>λ\right\}\\ \leq C_1\sharp\left\{k\in\mathbb{Z}:\sup_{n\geq 1}\left|\sum_{i\in \mathcal{B}_n} \!\!\raise{1.9ex}\hbox{$\scriptsize\prime$}\; \frac{a_{k+i}}{i}\right|>λ\right\},$$ for all $λ>0$, where $\mathcal{B}_n=\{-n, -(n-1), -(n-2),\dots , n-2, n-1, n\}$. Then there is a constant $C_2>0$ which does not depend on the sequence $\{a_j\}$ such that $$\sum_{n=1}^\infty\sharp\left\{k\in\mathbb{Z}:\left|\sum_{i=-n}^{n} \!\!\raise{1.9ex}\hbox{$\scriptsize\prime$}\; \frac{a_{k+i}}{i}\right|>λ\right\}\leq\frac{C_2}λ\sum_{i=-\infty}^{\infty}|a_i|$$ for all $λ>0$. Let $(X,\mathscr{B},μ)$ be a measure space, $τ:X\to X$ an invertible measure-preserving transformation, and suppose that $f\in L^1(X)$ such that for any sequence $(t_n)$ of integers there exists a constant $C_1>0$ such that $$μ\left\{ x: \sup_{n\geq 1}\left|\sum_{i\in \mathcal{B}_n-t_n}\!\!\!\raise{1.9ex}\hbox{$\scriptsize\prime$}\; \frac{f(τ^ix)}{i}\right| >λ\right\}\leq C_1μ\left\{x: \sup_{n\geq 1}\left|\sum_{i\in \mathcal{B}_n}\!\!\raise{1.9ex}\hbox{$\scriptsize\prime$}\; \frac{f(τ^i x)}{i}\right|>λ\right\} $$ for all $λ>0$, where $\mathcal{B}_n=\{-n, -(n-1), -(n-2),\dots , n-2, n-1, n\}$. Then there exists a constant $C_2>0$ which does not depend on $f$ such that $$\sum_{n=1}^\inftyμ\left\{x:\left|\sum_{i=-n}^{n}\!\!\raise{1.9ex}\hbox{$\scriptsize\prime$} \;\frac{f(τ^ix)}{i}\right|>λ\right\}\leq\frac{C_2}λ\|f\|_1$$ for all $λ>0$.
Let $K_n(x)$ denote the Fejér kernel given by $$K_n(x)=\sum_{j=-n}^n\left(1-\frac{|j|}{n+1}\right)e^{-ijx}$$ and let $σ_nf(x)=(K_n\ast f)(x)$, where as usual $f\ast g$ denotes the convolution of $f$ and $g$. Let the sequence $\{n_k\}$ be lacunary. Then the series $$\mathcal{G}f(x)=\sum_{k=1}^\infty \left(σ_{n_{k+1}}f(x)-σ_{n_k}f(x)\right)$$ converges unconditionally for all $f\in L^2(\mathbb{R})$. Let $(n_k)$ be a lacunary sequence, and $\{c_k\}_{k=1}^\infty \in \ell^\infty$. Define $$\mathcal{R}f(x)=\sum_{k=1}^\infty c_k\left(σ_{n_{k+1}}f(x)-σ_{n_k}f(x)\right).$$ Then there exists a constant $C>0$ such that $$\|\mathcal{R}f\|_2\leq C\|f\|_2$$ for all $f\in L^2(\mathbb{R})$, i.e., $\mathcal{R}f$ is of strong type $(2,2)$. As a special case it follows that $\mathcal{G}f$ also is of strong type $(2,2)$.
Let $(x_n)$ be a sequence and $ρ\geq 1$. For a fixed sequences $n_1 0$ such that $\|\mathcal{O}_ρ(ϕ_n\ast f)\|_{L^1(\mathbb{R})}\leq C\|f\|_{H^1(\mathbb{R})}$ for all $f\in H^1(\mathbb{R})$. (ii) Let $A_nf(x)=\frac{1}{n}\sum_{k=1}^nf(τ^kx)$ be the usual ergodic averages in ergodic theory. Then $\|\mathcal{O}_ρ(A_nf)\|_{L^1(X)}\leq C\|f\|_{H^1(X)}$ for all $f\in H^1(X)$. (iii) If $[f(x)\log (x)]^+$ is integrable, then $\mathcal{O}_ρ(A_nf)$ is integrable.
Let $f$ be a locally integrable function defined on $\mathbb{R}$, and let $(n_k)$ be a lacunary sequence. Define the operator $A_{n_k}$ by $$A_{n_k}f(x)=\frac{1}{n_k}\int_0^{n_k}f(x-t)\, dt.$$ We prove various types of new inequalities for the variation operator $$\mathcal{V}_sf(x)=\left(\sum_{k=1}^\infty|A_{n_k}f(x)-A_{n_{k-1}}f(x)|^s\right)^{1/s}$$ when $2\leq s<\infty$.
Let $T$ be an operator and suppose that there exists a positive constant $C$ such that $$\left(\int_I|Tf(x)|^q\, dx\right)^{1/q}\leq C\left(\int_I|f(x)|^q\, dx\right)^{1/q}$$ for every $q$ which is near enough to $1$ and for every interval $I$ in $\mathbb{R}$ and $f\in L^{\infty}(\mathbb{R})$. Then we show that $T$ maps $uL^{\infty}$ to ${\rm{BMO}}_u$.
We first prove that the well known transfer principle of A. P. Calderón can be extended to the vector-valued setting and then we apply this extension to vector-valued inequalities for the Hardy-Littlewood maximal function to prove the vector-valued strong type $L^p$ norm inequalities for $1<p<\infty$ and the vector-valued weak type $(1,1)$ inequality for ergodic maximal function.
Let $f$ be a measurable function defined on $\mathbb{R}$. For each $n\in\mathbb{Z}$ define the operator $A_n$ by $$A_nf(x)=\frac{1}{2^n}\int_x^{x+2^n}f(y)\, dy.$$ Consider the variation operator $$\mathcal{V}f(x)=\left(\sum_{n=-\infty}^\infty|A_nf(x)-A_{n-1}f(x)|^s\right)^{1/s}$$ for $2\leq s<\infty$. It has been proved in \cite{jkw1} that $\mathcal{V}$ is of strong type $(p,p)$ for $1<p<\infty$ and is of weak type $(1,1)$, it maps $L^\infty$ to BMO. We first provide a completely different proofs for these known results and in addition we prove that $\mathcal{V}$ maps $H^1$ to $L^1$. Furthermore, we prove that it satisfies vector-valued weighted strong type and weak type inequalities. As a special case it follows that $\mathcal{V}$ satisfies weighted strong type and weak type inequalities.