By Elias M. Stein
This ebook includes an exposition of a few of the most advancements of the final two decades within the following components of harmonic research: singular indispensable and pseudo-differential operators, the speculation of Hardy areas, L\sup\ estimates concerning oscillatory integrals and Fourier crucial operators, family members of curvature to maximal inequalities, and connections with research at the Heisenberg group.
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F m ( z l , . . , z n ) , z l , . . ,x,) = 0, i = l , ,. 224) in the neighborhood of Po. Note that if the Jacobi determinant [Eq. 120)] is zero at the point of interest, then we search for a different set of dependent variables to avoid the difficulty. 226) can be considered as a mapping from the xy space to the uu space. Under certain conditions, this maps a certain domain D,, in the xy space t o a certain domain D,, in the uu space on a one-to-one basis. Under such conditions, an inverse mapping should also exist.
If f ( x ) has derivative a t xo, it means that it is continuous a t that point. 42) 8 FUNCTIONAL ANALYSIS A geometric interpretation of the partial derivative is that the section of the surface z = f ( x , y ) with the plane y = yo is the curve z = f(x,yo); hence the partial derivative ~ ( X yo) O , is the slope of the tangent line (Fig. 2) t o z = f (x, yo) at ( 2 0 ,yo). Similarly, the partial derivative ~ ( x oyo) , is the slope of the tangent line to the curve z = f(x0, y) at (XO, yo). For a multivariate function the partial derivative with respect to the i t h independent variable is defined as df(X1,.
172) as dz = dz dz dX dY - dX + - dy. 175) This result can be extended t o any number of variables. In other words, any equation in differentials that is true in one set of independent variables is also true for another choice of variables. Formal proofs of these results can be found in books on advanced calculus (Apostol, Kaplan). 9 IMPLICIT FUNCTION THEOREM A function given as can be used to describe several functions of the form z = f(X,Y), y = g(x,z ) , etc. 181) both of which are defined in the domain x2 + y2 + z 2 5 9.