Is program analysis part of programming language theory? If I am correct, programming language theory studies 


syntax and semantics of programming languages, and 
design and implementation of programming languages (e.g. compiler, ...)

Does programming in programming languages, i.e. creating programs in
programming languages, belong to programming language theory?
Does program analysis (correctness and performance of programs)
belong to programming language theory? My thought is:


semantics of a programming language is used for verifying correctness of programs in the language, so it seems like the answer
is yes, although I am not sure if analysis of programs' performance
belongs to programming language theory.
program analysis also seems to belong to software engineering, which makes me feel perhaps it doesn't belong to programming
language theory.



Thanks.
"Programming language theory" more or less includes everything you have listed, but of course there is overlap with other areas, such as software engineering, algorithms, security, human-computer interaction, and good old mathematics.

For instance, if you look at last year's programme of the Principles of Programming Languages (POPL) conference you will see not only topics about design, implementation and semantics of programming languages, but also about security, program analysis, papers about resource complexity, and many others.

In general I would say it good to be open about what "belongs" to a "research area". It is a shame to miss opportunities for new discoveries because of narrow-mindedness.