In this work, we study the continuity of pseudodifferential operators on local Hardy spaces h p (R n ) and generalize the results due to Goldberg and Taylor by showing that operators with symbols in S 0 1,δ (R n ), 0 ≤ δ < 1, and in some subclasses of S 0 1,1 (R n ) are bounded on h p (R n ) (0 < p ≤ 1). As an application, we study the local solvability of the planar vector field L = ∂ t + ib(x, t)∂ x , b(x, t) ≥ 0, in spaces of mixed norm involving Hardy spaces.

It is well known that, for space dimension n > 3, one cannot generally expect L 1 -L p estimates for the solution ofwhere (t, x) ∈ R + × R n . In this paper, we investigate the benefits in the range of 1 ≤ p ≤ q such that L p -L q estimates hold under the assumption of radial initial data. In the particular case of odd space dimension, we prove L 1 -L q estimates for 1 ≤ q < 2n n−1 and apply these estimates to study the global existence of small data solutions to the semilinear wave equation with power nonlinearity |u| σ , σ > σ c (n), where the critical exponent σ c (n) is the Strauss index.Keywords Wave equation · L 1 estimates · Asymptotic behavior of solutions · Critical exponent · Global existence of small data solutions Mathematics Subject Classification 35L05 (primary) · 35B33 · 35B40 · 74G25 M. R. Ebert and T. Picon are partially supported by São Paulo Research Fundation (Fapesp) Grants 2013/20297-8 and 2013/17636-5, respectively.

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Keywords:Quasilinear weakly hyperbolic operators Levi conditions Loss of regularity in the Sobolev scale Energy methodIn this paper we determine bounds for the optimal loss of regularity in the Sobolev scale for a class of weakly hyperbolic operators.

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