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We establish an LP a priori estimate for the Tricomi equation. Our main tool is Mihlin's multiplier theorem combined with well-known estimates of the Newtonian potential. 0. INTRODUCTION The purpose of this paper is to establish an LP a priori estimate for the Tricomi equation in the upper half-space. The Tricomi equation arises in transonic gas dynamics, and is a typical model equation of changing type. It has been extensively investigated from the various viewpoints. The Tricomi equation can be interpreted as an elliptic equation which degenerates on the boundary, which is our viewpoint in this paper. We can formulate the Dirichlet boundary value problem in the upper half-space as follows. 02 + Yacuz=f, in R+ (0.2) u(0,x) = (x), on MT, where R+ {(y, x) : y > 0, x E Rn-} and /x is the Laplacian in the x variable. An L2 a priori estimate can be obtained very easily. Following the presentation in [8], we suppose that u E Co (Rn), and multiply the equation (0.1) by &92U Then, we integrate over R,r to obtain (0.3) IR ( dy2x)2Jj+ V dydx Rn-1 o 9Y2 Rn-1 o a'y 2IRn1l |Vq$ dx + j-1 f ,3 dydx, where Vx stands for the gradient in x E R n-. This yields (0.4) 2U + \F 1 +YX 4Y21 L2(Rn) Dy L2(Rn) L2(Rn) < M(||f 1|L2(Rn) + 11VX01iL2(Rn-1))7 Received by the editors December 30, 1996 and, in revised form, February 10, 1998. 1991 Mathematics Subject Classification. Primary 35J70, 35B45.

An $L^p$ a priori estimate for the Tricomi equation in the upper half space