We study the bi-commutators $[T_1, [b, T_2]]$ of pointwise multiplication and Calder\'on-Zygmund operators, characterize their $L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}$ boundedness for several off-diagonal regimes mixed-norm integrability exponents $(p_1,p_2)\neq(q_1,q_2)$. The strategy is based on a bi-parameter version recent approximate weak factorization method.

Let $[\, \cdot\,]$ be the floor function. In this paper, we prove that when $1<c<\frac{16559}{15276}$, then every sufficiently large positive integer $N$ can represented in form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3]\,, \end{equation*} where $p_1,p_2,p_3$ are primes, such $p_1=x^2 + y^2 +1$.

We establish convergence in norm and pointwise almost everywhere for the non-conventional (in sense of Furstenberg) bilinear polynomial ergodic averages $$ A_N(f,g)(x) := \frac{1}{N} \sum_{n=1}^{N} f(T^nx) g(T^{P(n)}x) as $N \to \infty$, where $T\colon X\to X$ is a measure-preserving transformation $\sigma$-finite measure space $(X,\mu), P(\mathrm{n})\in \mathbb{Z}[\mathrm{n}]$ degree $d \geq 2...

Let $${{\mathfrak {A}}}\, $$ and '$$ be two $$C^*$$ -algebras with identities $$I_{{{\mathfrak }$$ '}$$ , respectively, $$P_1$$ $$P_2 = I_{{{\mathfrak } - P_1$$ nontrivial projections in . In this paper, we study the characterization of multiplicative $$*$$ -Lie–Jordan-type maps, where notion these maps arise here. particular, if $${\mathcal {M}}_{{{\mathfrak is a von Neumann algebra relative $...

We study the persistent homology of an Erdős–Rényi random clique complex filtration on n vertices. Here, each edge e appears independently at a uniform time $$p_e \in [0,1]$$ , and persistence cycle $$\sigma $$ is defined as $$p_2(\sigma ) / p_1(\sigma )$$ where $$p_1(\sigma are birth death times . show that if $$k \ge 1$$ fixed, then with high probability maximal k-cycle order $$n^{1/k(k+1)}$$

We obtain the criterion of best approximant for multivariable functions in $L_{p_1,\ldots p_n}$ space.

It is shown that every measurable partition ${A_1,..., A_k}$ of $\mathbb{R}^3$ satisfies $$\sum_{i=1}^k||\int_{A_i} xe^{-\frac12||x||_2^2}dx||_2^2\le 9\pi^2.\qquad(*)$$ Let ${P_1,P_2,P_3}$ be the partition of $\mathbb{R}^2$ into $120^\circ$ sectors centered at the origin. The bound is sharp, with equality holding if $A_i=P_i\times \mathbb{R}$ for $i\in {1,2,3}$ and $A_i=\emptyset$ for $i\in \{4...

The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim solve functional $$f^{-1}\Bigl(\frac{p_1(x)f(x) +p_2(y)f(y)}{p_1(x)+p_2(y)}\Bigr)+g^{-1}\Bigl(\frac{q_1(x)g(x) +q_2(y)g(y)}{q_1(x)+q_2(y)}\Bigr)=x + y \ (x,y\in I),$$ where I a nonempty open real interval and $$f,g \colon \to\mathbb{R}$$ are contin...