It is well known that standard Calderón-Zygmund singular integral operators with isotropic and nonisotropic homogeneities are bounded on the classical H p (R m ) and nonisotropic H p h (R m ), respectively. In this paper, we develop a new Hardy space theory and prove that the composition of two Calderón-Zygmund singular integral operators with different homogeneities is bounded on this new Hardy space. Such a Hardy space has a multiparameter structure associated with the underlying mixed homogeneities arising from the two singular integral operators under consideration. The Calderón-Zygmund decomposition and an interpolation theorem hold on these new Hardy spaces.Mathematics Subject Classification (2010): Primary 42B30; Secondary 42B20.

In this paper, we are concerned with the local existence and singularity structure of low regularity solutions to the semilinear generalized Tricomi equation ∂ 2 t u − t m ∆u = f (t, x, u) with typical discontinuous initial data (u(0, x), ∂ t u(0, x)) = (0, ϕ(x)); here m ∈ N, x = (x 1 , ..., x n ), n ≥ 2, and f (t, x, u) is C ∞ smooth in its arguments. When the initial data ϕ(x) is a homogeneous function of degree zero or a piecewise smooth function singular along the hyperplane {t = x 1 = 0}, it is shown that the local solutionand is C ∞ away from the forward cuspidal cone Γ 0 = (t, x): t > 0, |x| 2 = 4t m + 2 , respectively. On the other hand, for n = 2 and piecewise smooth initial data ϕ(x) singular along the two straight lines {t = x 1 = 0} and {t = x 2 = 0}, we establish the local existence of a solution2 )) in general due to the degenerate character of the equation under study; here Γ. This is an essential difference to the well-known result for

We prove the local existence and uniqueness of minimal regularity solutions u of the semilinear generalized Tricomi equationunder the assumption that |F (u)| |u| κ and |F ′ (u)| |u| κ−1 for some κ > 1. Our results improve previous results of M. Beals [2] and of ourselves [15][16][17]. We establish Strichartz-type estimates for the linear generalized Tricomi operator ∂ 2 t − t m ∆ from which the semilinear results are derived.

This paper is a continuation of our previous work [21], where we have established that, for the secondorder degenerate hyperbolic equation ∂ 2 t − t m ∆x u = f (t, x, u), locally bounded, piecewise smooth solutions u(t, x) exist when the initial data (u, ∂tu) (0, x) belongs to suitable conormal classes. In the present paper, we will study low regularity solutions of higher-order degenerate hyperbolic equations in the category of discontinuous and even unbounded functions. More specifically, we are concerned with the local existence and singularity structure of low regularity solutions of the higher-order degenerate hyperbolic equations ∂t ∂ 2, respectively; here m, m 1 , m 2 ∈ N, m 1 = m 2 , x ∈ R n , n ≥ 2, and f is C ∞ smooth in its arguments. When the ϕ i and ψ j are piecewise smooth with respect to the hyperplane {x 1 = 0} at t = 0, we show that local solutionsare two characteristic surfaces forming a cusp. When the ϕ i and ψ j belong to C ∞ 0 (R n \ {0}) and are homogeneous of degree zero close to x = 0, then there exist local solutionsrespectively; here Γ k = (t, x) : t ≥ 0, |x| 2 = 4t k+2 (k + 2) 2 (k = m, m 1 , m 2 ) is a cuspidal conic surface ("forward light cone") and l 0 = {(t, x) : t ≥ 0, |x| = 0} is a ray.

In [20,21], we have established the existence and singularity structures of low regularity solutions to the semilinear generalized Tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations, respectively. In the present paper, we shall be concerned with the low regularity solution problem for the semilinear mixed type equation f (t, x, u) is C 1 smooth in its arguments and has compact support with respect to the variable x. Under the assumption of the subcritical growth of f (t, x, u) on u, we will show the existence and regularity of the considered solution in the mixed type domain [−T 0 , T 0 ] × R n for some fixed constant T 0 > 0.