For instance, if "P"("x",ξ) is an infinitely differentiable function on R"n" × R"n" with the property  for all "x",ξ ∈R"n", all multiindices α,β, some constants "C"α, β and some real number "m", then "P" belongs to the symbol class formula_18 of Hörmander. The corresponding operator "P"("x","D") is called a pseudo-differential operator of order m and belongs to the class formula_19  Linear differential operators of order m with smooth bounded coefficients are pseudo-differential operators of order "m". The composition "PQ" of two pseudo-differential operators "P", "Q" is again a pseudo-differential operator and the symbol of "PQ" can be calculated by using the symbols of "P" and "Q". The adjoint and transpose of a pseudo-differential operator is a pseudo-differential operator. If a differential operator of order "m" is (uniformly) elliptic (of order "m") and invertible, then its inverse is a pseudo-differential operator of order −"m", and its symbol can be calculated. This means that one can solve linear elliptic differential equations more or less explicitly by using the theory of pseudo-differential operators. Differential operators are "local" in the sense that one only needs the value of a function in a neighbourhood of a point to determine the effect of the operator. Pseudo-differential operators are "pseudo-local", which means informally that when applied to a distribution they do not create a singularity at points where the distribution was already smooth. Just as a differential operator can be expressed in terms of "D" = −id/d"x" in the form  for a polynomial "p" in "D" (which is called the "symbol"), a pseudo-differential operator has a symbol in a more general class of functions. Often one can reduce a problem in analysis of pseudo-differential operators to a sequence of algebraic problems involving their symbols, and this is the essence of microlocal analysis. Pseudo-differential operators can be represented by kernels. The singularity of the kernel on the diagonal depends on the degree of the corresponding operator. In fact, if the symbol satisfies the above differential inequalities with m ≤ 0, it can be shown that the kernel is a singular integral kernel.