We define and characterize the harmonic Besov space Bp, 1 ≤ p ≤ ∞, on the unit ball B in Rn. We prove that the Besov spaces Bp, 1 ≤ p ≤ ∞, are natural quotient spaces of certain Lp spaces. The dual of Bp, 1 ≤ p < ∞, can be identified with Bq , 1/p + 1/q = 1, and the dual of the little harmonic Bloch space B0 is B1.

In this paper, we consider the [Formula: see text]-Hardy inequalities on the sphere. By the divergence theorem, we establish the [Formula: see text]-Hardy inequalities on the sphere. Furthermore, we also obtain their best constants. Our results can be regarded as the extension of Xiao's (J. Math. Inequal. 10:793-805, 2016).

The purpose of this paper is to link informations on the application u 7→ gu with some growth conditions on the functions u and g subharmonic in the unit ball of R . Two kinds of growth are considered: the Bloch–type growth and growth conditions expressed through integrals involving involutions of the unit ball.

1.2 The theorem 1.1 contrasts the rank one case, where the secondary class of the flat bundle, associated to a representation ρ : π1(X) → GL1(C) = C ∗ is essentially the representation itself, so it may be completely arbitrary. But this freedom becomes limited when one looks at the line bundles over curves, corresponding via the Deligne-Ramakrishnan construction to elements of K2(X). The map r ...

We describe a construction of an object associated to the fundamental group of P−{0, 1,∞} in the Bloch-Kriz category of mixed Tate motives. This description involves Massey products of Steinberg symbols in the motivic cohomology of the ground field.

Recommended by Radu Precup The boundedness and compactness of the weighted composition operators from logarithmic Bloch-type spaces to Bloch-type spaces are studied here.