In this paper, we study maximal and square functions associated with bilinear Bochner–Riesz means at the critical index. particular, prove that they satisfy weighted estimates from \(L^{p_1}(w_1)\times L^{p_2}(w_2)\rightarrow L^p(v_w)\) for weights \((w_1,w_2)\in A_{\textbf{P}}\) where \(p_1,p_2>1\) \(\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}\). Also, show both operators fail to weak-type end-poi...

In this paper we obtain order estimates for the Kolmogorov widths of set $B_{p_0}^m\cap \nu B_{p_1}^m$ in $l_q^m$; here $0\le n\le m/2$, $p_0>p_1$, $q<\infty$.

We consider the solutions to inequality \[|p_1^c + \cdots p_s^c - R| < R^{-\eta}\] (where $c > 1$, \not\in \mb N$ and $\eta$ is a small positive number; $R$ large). obtain new ranges of $c$ for which this has many in primes $p_1, \ldots, p_s$, $s = 2$ (and `almost all' $R$), $s=3$, 4 and~5. also equation integer parts \[[p_1^c] [p_s^c] r\] where $r$ large. Again $c> $c\not\in N$. primes, $s=3$ 5.

Abstract Given finite configurations $P_1, \dots , P_n \subset \mathbb {R}^d$ let us denote by $\mathbf {m}_{\mathbb {R}^d}(P_1, P_n)$ the maximum density a set $A \subseteq can have without containing congruent copies of any $P_i$ . We will initiate study this geometrical parameter, called independence considered configurations, and give several results we believe are interesting. For instance...

Let [ ⋅ ] be the floor function. In this paper, we show that every sufficiently large positive integer N can represented in form $$\displaystyle N=[p_1\log p_1]+[p_2\log p_2]+[p_3\log p_3], $$ where p1, p2, andp3 are prime numbers. We also establish an asymptotic formula for number of such representations, when do not exceed given number.

We study traces of weighted Triebel–Lizorkin spaces F p , q s ( R n w ) $F^s_{p,q}(\mathbb {R}^n,w)$ on hyperplanes − k $\mathbb {R}^{n-k}$ where the weight is Muckenhoupt type. concentrate example α x = | $w_\alpha (x) {\big\vert x_n\big\vert }^\alpha$ when ≤ 1 $\big\vert \le 1$ ∈ $x\in \mathbb {R}^n$ and (x)=1$ otherwise, > $\alpha >-1$ . Here we use some refined atomic decomposition argument...

The classic string indexing problem is to preprocess a S into compact data structure that supports efficient pattern matching queries. Typical queries include existential (decide if the occurs in S), reporting (return all positions where occurs), and counting number of occurrences pattern). In this paper we consider variant indexing, goal compactly represent such given two patterns $$P_1$$ $$P_...

Let $$\mathfrak X$$ be the homogeneous tree of degree $$q+1~(q\ge 2)$$ . In this article, we present symbolic criteria for boundedness pseudo differential operators on A sufficient condition weak type (p, q) is also given. We necessary and conditions symbols $$\sigma $$ such that corresponding pseudo-differential $$T_\sigma from $$L^{p_1} (\mathfrak X)$$ into $$L^{p_2} to nuclear $$1\le p_1, p_...