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.

We study the class of all finite directed graphs up to primitive positive constructability. The resulting order has a unique greatest element, namely graph $P_1$ with one vertex and no edges. lower bound, $P_2$ two vertices edge. Our main result is complete description bounds $P_2$; we call these submaximal. show that every not equivalent below submaximal graphs.

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_...

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...

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...

We study the complexity of reasoning in abstracts argumentation frameworks close to a graph classes that allow for efficient reasoning methods, i.e. to one of the classes of acyclic, noeven, biparite and symmetric AFs. In this work we show that certain reasoning problems on the second level of the polynomial hierarchy still maintain their full complexity when restricted to instances of fixed di...

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_...